3.e4 is more theoretically burdening. Black plays 3...Nc6 sometimes, and usually White gambits a pawn in the resulting complications. White needs to know his stuff.
After the main line 3...e5, White needs to know a lot of theory to get an edge.
3.Nf3 is easier because White has a clear plan: e3, Bxc4, castle, and it’s an easy middlegame game to play.
It should be mentioned that 3.e3 is an inaccurate move order, as Black has the additional option of 3...e5!?
"3. e3 This is the other main move, and I don't really understand it."
It is simple: the black c pawn is attacked by f1 Bishop. The try to defend it with 3...b5 falls into a trap after 4.a4 c6 5.axb5 cxb5 6.Qf3!
This for Joseph too (3.e3 is an inaccurate move order) 
After 3...e5 4.Bxc4 exd4 5.exd4 White is better.
An important theorical game, which caused some rebirth for e3 at high level :
I don't know the theorical status nowdays but the plan with a6-b5-Bb7 was supposed to gives easy equality for black.
White had to find other ways, like 3.e4 or 3.e3 (Aronian opted for e3 after this game so the rebirth of this move).
Not sure 1.d4 d5 2.c4 dxc4 3.e3 e5 is so easy for black, giving a good IQP for white.
Razuvayev made a superb chapter on it called "U were right M.La Bourdonnais!" (who played e3!), in a Dvoretsky's book.
"3. e3 This is the other main move, and I don't really understand it."
It is simple: the black c pawn is attacked by f1 Bishop. The try to defend it with 3...b5 falls into a trap after 4.a4 c6 5.axb5 cxb5 6.Qf3!
This for Joseph too (3.e3 is an inaccurate move order) After 3...e5 4.Bxc4 exd4 5.exd4 White is better.
Actually, it’s not clear that white is better after 5.exd4. He has an initiative, but with accurate play, White doesn’t actually get an advantage. Play through some games and you’ll see what I mean. If Black plays accurately White has nothing.
3. e4 is a good move...but its more messy. Lots of queens gambit players don't really like to get involved in highly tactical battles like that. 3 nf3 also does well enough...and its less effort...not as much tactics to calculate.
AFAIK, there are some lines where you want to play Qf3 and attack on the a8 (diagonal), but you don't have this threat on the Rook when you've played e4. In general, I think what @Vercingetorix75 said is true
e3 is more solid - e4 is more aggresive
I like the early b5 lines for black...
The 'trap' that white can spring in the e3 line doesn't have to be shut...
Instead of c6 black can play b4. After develop e6/Nf6/Bb7/Nbd7/0-0 and prepare c5
After 1d4 d5 2c4 dc 3Nf3 Nf6 4e3 b5 with similar play... I like blacks position, it reminds me of the Kan Sicilian...
After 1d4 d5 2c4 dc 3e4 black has an interesting exchange sacrifice, with pawn compensation and asymmetry. My engine doesn't give more than a usual slight plus for white, but the first player may feel over confident, feeling perhaps that black has blundered...
10.f3 and White is over 1 pawn up (already). In the 3.e3 variation, the position is still determined by 5.Qf3 that must be followed by c6, or lose the Rook.
in this variation, 9...e6! is the critical move, not Bxe4.
The idea is to wait a move to take e4, like in this game :
White has improvments in this game but the main move is of course 10.f3, then black has choice : Nc6, Nf6, or the risky f5!
Things are really difficult to assess here.
10.f3 and White is over 1 pawn up (already). In the 3.e3 variation, the position is still determined by 5.Qf3 that must be followed by c6, or lose the Rook.
I think this is still playable for black...
Poucin, as you say 10f3 is best for white, according to my engine, and gives white an 'over 1' advantage. Enginewise, taking the pawn with Bxe4 still gives ~0.7 + for white. Who knows. Perhaps a good choice for blitz etc :-)
On reflection I'm more confident with b5 against non-2e4 lines than with them. Perhaps this shows that for white 1d4 d5 2c4 dc 3e4 is best. This is reflected by the stats and GM Chris Ward's assertion that 3e4 might(should?) be considered the main line...
If you want to know how to play the QGA as Black, study Matthew Sadler's games. Okay, the theory of 3 e4 may have moved on a bit since then, but his play in the 3 Nf3 main lines is still exemplary.
Some players also prefer to play 2.Nf3 as a move-order to avoid a couple of openings (including against 1....Nf6, so you could have games starting 1.d4 Nf6 2.Nf3 d5 3.c4 dxc4 as well), which limits their choice.
3.e3 was desinged not only as a trap, but also as a way to avoid the ....Bg4 variation of the QGA.
Just to add some more context: historically, it was always known that ...e5 was the equalizing break in the QGA, similar to how ...d5 was the main break against the King's Gambit. If Black can get this move in, he generally does well. On the same token, if you can restrict this break, then Black can become cramped and without any active plans.
If White plays the 3.e4 lines, Black can strike back with the break immediately, 3...e5, and reach a nice position. Historically, the line was considered equal, as neither dxe5 nor d5 gave White anything. The classical approach then came up with a different idea, 3.Nf3, stopping the break and making Black find a different plan. If Black does nothing, White will simply consolidate, push e4 when his position is perfect and have a big advantage.
The modern idea of playing 4.Nf3 in the 3.e4 e5 variation and often gambiting the d4-pawn was not explored until relatively recently, and it's quite different from the historical QGA, for both sides. I don't know if it is 'better' than any other variation of the QGA, but it makes both sides solve concrete puzzles immediately, and a single slip can be fatal (again for both sides). For the practical tournament player, that can be a useful thing.
The modern idea of playing 4.Nf3 in the 3.e4 e5 variation and often gambiting the d4-pawn was not explored until relatively recently, and it's quite different from the historical QGA, for both sides. I don't know if it is 'better' than any other variation of the QGA, but it makes both sides solve concrete puzzles immediately, and a single slip can be fatal (again for both sides). For the practical tournament player, that can be a useful thing.
That's interesting. I have always overlooked that white was actually gambiting here ! 
Some players also prefer to play 2.Nf3 as a move-order to avoid a couple of openings (including against 1....Nf6, so you could have games starting 1.d4 Nf6 2.Nf3 d5 3.c4 dxc4 as well), which limits their choice.
Good point. Of course white can enter from 1Nf3 d5 - possibly the best move 2d4 - white feels he has 'tricked' black into a queens pawn opening 2...Nf6 - black says 'I'm not afraid of the London'. 3c4 - white feels vindicated. 3...dc - black says 'You haven't got the most pressing variation of the QGA. Tis I who feel vindicated !'
Just to add some more context: historically, it was always known that ...e5 was the equalizing break in the QGA, similar to how ...d5 was the main break against the King's Gambit. If Black can get this move in, he generally does well. On the same token, if you can restrict this break, then Black can become cramped and without any active plans.
If White plays the 3.e4 lines, Black can strike back with the break immediately, 3...e5, and reach a nice position. Historically, the line was considered equal, as neither dxe5 nor d5 gave White anything. The classical approach then came up with a different idea, 3.Nf3, stopping the break and making Black find a different plan. If Black does nothing, White will simply consolidate, push e4 when his position is perfect and have a big advantage.
The modern idea of playing 4.Nf3 in the 3.e4 e5 variation and often gambiting the d4-pawn was not explored until relatively recently, and it's quite different from the historical QGA, for both sides. I don't know if it is 'better' than any other variation of the QGA, but it makes both sides solve concrete puzzles immediately, and a single slip can be fatal (again for both sides). For the practical tournament player, that can be a useful thing.
All this is like i say so many times : abstract...
Times changed, and the e5 break is proved nowdays to be insufficient to claim equality for black
In a Dvoretsky book "opening preparation", a chapter written by Razuvayev with the nice title : "u were right M. La Bourdonnais", deals with this e5 break (in the line 1.d4 d5 2.c4 dxc4 3.e3 e5).
At the time the matches La Bourdonnais-McDonnell were played, it was thought that this break was enough for equality but La Bourdonnais crushed McDonnell with the IQP positions resulting.
Razuvayev point is that : La Bourdonnais showed that the dogmatic point of view that e5 is good for black is not at all true.
Theory agreed almost 200 years later with La Bourdonnais.
When the book was written (2009), we didnt know (i think) the line played several times by Svidler (given above), which casts a serious doubt about 3.Nf3. Then it was the rebirth of 3.e3 which gave reason to Razuvayed and La Bourdonnais.
So ok, follow principles, but take care about dogms and concepts : they can be ok and useful, but chess is a concrete game.
A game played today :
3.Nf3 is just enough to get a draw...
Thanks for the comments, everyone. So 3. e4 is theoretically a better way to play for an advantage if you know the lines, but if you don't is more likely to get you into trouble, whereas 3. Nf3 or 3. e3 gets you into a playable middlegame, and 2. Nf3 (deferring c4) is an option too.
Why is 3. Nc3 so rarely played? Stockfish rates 3. Nf3 on average at +0.4 to 0.5 while 3. Nc3 has a broader range of +0.27 to 0.51. Immediate recapture on c4 is not essential as 3...c5 leads to a gunsberg variation where after 4. d5 you can reclaim after 4...Nf6. Stockfish also likes 3...a6 but after 4. e3 b5 5. a4 allows for some good counterplay.
Why is 3. Nc3 so rarely played? Stockfish rates 3. Nf3 on average at +0.4 to 0.5 while 3. Nc3 has a broader range of +0.27 to 0.51. Immediate recapture on c4 is not essential as 3...c5 leads to a gunsberg variation where after 4. d5 you can reclaim after 4...Nf6. Stockfish also likes 3...a6 but after 4. e3 b5 5. a4 allows for some good counterplay.
Well, i mentionned lines where black can play a6 to prepare b5. This is an effective idea against 3.Nf3, and more against 3.Nc3.
Some points :
I don't understand many of your analysis vicavox (like your 3.Nc3 e5 4.Nf3?! when 4.e3 is the move, going for an IQP position, as i discussed some posts above).
I guess that is because your analysis are only lines given by engines, which don't know theory and cannot discover or find the truth so early. To work an opening, u have to use engine but also sources (books and others), to help engine and to give it some directions. An engine is useless in opening preparation if u use it all alone.
In the Queen's Gambit Accepted, according to the database, white's most common third move by quite some margin is 3. Nf3: https://www.chess.com/explorer?moveList=d4+d5+c4+dxc4&ply=4
Why is this much more common than 3. e4, which puts another pawn in the center, threatens to take back the c4 pawn, and scores notably better?