2... f6 is not bad. Andd if 3.Bh4 then Nh6-f5
d4 d5 Bg5

2...f6 is supposed to be Black's best reply, which wasn't thought to be the case 10 years ago, but a recent book on the Trompowsky, predominantly written for White, actually recommends this line for Black!

After 1..f6 I think 2. Bd2 is playable. Although it's not agressive it's make f6 look akward. Overall it looks like a great surprise weapon espcially if you only need a draw to win a tournament.

To be completely honest, really the only advantages it has over the QG are that there isn't much theory on it (which allows some creativity), and opponents are usually very unfamiliar with it. That's basically it. It's certainly sound, but the Queen's Gambit is a much better try for an advantage. Many people compare it to the Trompowsky, but the key difference is that Black hasn't exposed his/her knight yet, so Black can gain a couple tempos by chasing it around, and after 2...h6 3. Bh4 c6, Black has a solid position. Of course, Black can always transpose back to the Trompowsky with 2...Nf6, but really it's better to go with the line I just mentioned. If you're interested in it because you like to be creative and don't like theory, go ahead, it's a decent line. If you're looking at it because you want an advantage, stop immediately and go look at something else.
After 1. d4 d5 2. Bg5 h6 3. Bh4 c6 4. e3 Qb6. Do you prefer 5. Qc1 or 5.b3?

What advantages are there to this over Queen's Gambit?
None.
No there is the avoid theory advantage and it is not better for Black. Maybe equal but nothing more.

@FirePlanet10: I prefer 5. Qc1 myself, I don't really trust b3 quite as much. My personal thinking (which probably is a bit flawed) is that it keeps some options open while leaving the queenside pawns a bit flexible.
@ThrillerFan & @pfren: Yeah, 2...f6 is supposed to be best and it's the better test of the Levitsky, I just stuck in my earlier line because it's also known to be solid and what I play myself (although I'm thinking about switching to 2...f6). And the book ThrillerFan is wondering about is Richard Pert's Playing the Trompowsky, which also has chapters on the related Levitsky and Anti-Dutch. Pert does admit that 2...f6 is the best way to play against it and that it can be a bit tricky for White, although I don't have the book with me right now to look over the exact lines. As for 2...Nf6, I'm personally fine with playing against it as White whenever I play the Levitsky/Trompowsky, but it's definitely still a solid choice.
@rdecredito: I'm fine with playing 4.dxc5 Qa5+ 5. c3 Qxc5 6. Nd2 with a decent position against 3...c5. Although yeah, if you don't really count the "avoiding the QG's massive amount of theory" as an advantage, then the Levitsky doesn't really have anything over the Queen's Gambit.
To sum it up: It's an opening that doesn't give an advantage but is a good surprise opening to avoid theory and usually the better player will win. Also like I said before I could just use it to draw if I just need it to win a tournament.

every time white doesnt play 2c4, I just play 2... c5 as black(well not against 2e4 and i guess not against 2 b4 but you know what i mean). All the unusual q pawn games are somewhat too slow to punish black for that.And it makes the central contest much easier.
d4 d5 Bg5 c5?! is inaccurate since White could play dxc5

This opening tends to be very drawish when I have played it but I could use it against lower rated opponents.

every time white doesnt play 2c4, I just play 2... c5 as black(well not against 2e4 and i guess not against 2 b4 but you know what i mean). All the unusual q pawn games are somewhat too slow to punish black for that.And it makes the central contest much easier.
d4 d5 Bg5 c5?! is inaccurate since White could play dxc5
.. and give up control of the center, therefore all hope for an advantage, for no reason.
c5 has only been played 250 games and has 45% win for White. And no black doesn't get an advantage. Better for black is either 2...f6, c6 or h6

White doesn't have to take the Knight just play 3.e3 and it's fine for White.
I assume you mean 4.e3.
Well, in that case how do you justify the move 2.Bg5 after the natural 4...Ne4? Black is already a bit better.
By playing 2.Bg5 and then not taking on f6, you rather admit that you were bluffing.
I mean after 1. d4 d5 2.Bg5 Nf6 3.e3 Ne4 4.Bf4 Tranposes to a line of the Trompowsky which Black is not better in. Oh but after 1.d4 d5 2.Bg5 c5 3.dxc5 Nf6 then yes 4. Bxf6 but Black is not better. The opening in general is a surprise weapon. White can get an advantage if Black messes up.
What advantages are there to this over Queen's Gambit?