I'm not sure what your question is (you think the other line is better or worse?)
The point is black starts the puzzle up a piece, so by making an even trade you stay ahead. Stuff you're saying about the a file doesn't matter since black is up a whole piece.
but black is only up a knight and bishop on the move the puzzle ends. They lose the bishop the very next move, so they'd only be up a knight, which is the same with the line I had in mind. So that's why I don't understand why the puzzle line is better than the line I had in mind. There's still the same trades being made, but white's board is worse.
but black is only up a knight and bishop on the move the puzzle ends. They lose the bishop the very next move, so they'd only be up a knight, which is the same with the line I had in mind. So that's why I don't understand why the puzzle line is better than the line I had in mind. There's still the same trades being made, but white's board is worse.
They are more or less equal, although the move Qb3+ will make Black think: am I going to lose a piece back? It's kind of a more challenging response to consider.
I'm sure there are players who failed the puzzle after Qb3+.
I don't understand why the puzzle line is better than the line I had in mind [since the material is the same either way]
Sure, both lines are equally good.
The engine probably prefers the puzzle line since the white king can still castle and the rook is opened on the a file... but the difference is very small, the lines are equal but it can't give both as a solution.
... but mostly... this is not a useful question 
It's like walking through the wreckage of a house that burned down, and noting that the home owners could have potentially saved a little money by using a different shade of paint, when their house burned down because they enjoyed setting off fireworks inside their home... Black is up a piece. Being up a piece is massive. Whether or not the a file is open isn't even a discussion... but if it were, sure, you could argue it's VERY SLIGHTLY better to suffer the doubled pawn in order to have the a file open for white's rook, so chess.com's line is better.
https://www.chess.com/puzzles/problem/1110500
https://www.chess.com/analysis?fen=r2q1bnr/pp1n1kpp/2pp4/8/3PP1b1/5N2/PP3PPP/RNBQK2R%20w%20KQ%20-%200%208&flip=true
You are Black. Solution is: Ng5+, Qxg5, Qb3+, Be6, Bxg5, Bxe6. The puzzle stops at Bxe6 but as white, axb3 seems the natural next move, which opens the a file for the rook.
Now back to the beginning...I was expecting and looking to play this line: Ng5+, Qxg5, Bxg5, Bxd1, Kxd1. There is no difference in material (black still loses the queen and bishop on the move after the puzzle, and white still loses the knight and queen), but now the a file rook is still pinned, and the king cannot castle if it takes on d1. If king doesn't take on d1, then you're up a bishop.
If you were White...would you castle or take the bishop on d1? The queens are off the board, so castling might not be of much value by then.
https://www.chess.com/puzzles/problem/1110500
https://www.chess.com/analysis?fen=r2q1bnr/pp1n1kpp/2pp4/8/3PP1b1/5N2/PP3PPP/RNBQK2R%20w%20KQ%20-%200%208&flip=true
You are Black. Solution is: Ng5+, Qxg5, Qb3+, Be6, Bxg5, Bxe6. The puzzle stops at Bxe6 but as white, axb3 seems the natural next move, which opens the a file for the rook.
Now back to the beginning...I was expecting and looking to play this line: Ng5+, Qxg5, Bxg5, Bxd1, Kxd1. There is no difference in material (black still loses the queen and bishop on the move after the puzzle, and white still loses the knight and queen), but now the a file rook is still pinned, and the king cannot castle if it takes on d1. If king doesn't take on d1, then you're up a bishop.
If you were White...would you castle or take the bishop on d1? The queens are off the board, so castling might not be of much value by then.
100% capture. It's the natural move in that position, which is why it was the last move in my line. But....in the puzzle line, the king can still castle, which is why I made a note of it since white would then have 2 open rooks and king protection.
In a chess.com puzzle it is none of your business how the opposite side defends - good, bad or ugly. Your only responsibility is to play the best moves for the side you are on which is black here. The counter play is chosen to provide you with the best learning experience on the rating level where you are and thus improve your solving skill.
White is lost either way and the puzzle is made for black so white will make mistakes. Best play from both sides is probably this:
Notice how Qb3+ avoids your line completely and white will still be able to castle regardless of what you do. It is also check which is the most forcing. Forcing moves are very important and limit your opponents options. When calculating your lines, you have to see these moves as they are the most important.
But from your given line, it does seem that you take the weakness of doubled pawns too lightly.
In a chess.com puzzle it is none of your business how the opposite side defends - good, bad or ugly. Your only responsibility is to play the best moves for the side you are on which is black here. The counter play is chosen to provide you with the best learning experience on the rating level where you are and thus improve your solving skill.
Meh... there is definitely learning to be had by studying both sides of puzzles. It all depends on if YOU understand what is happening or not. There is nothing wrong with trying to understand something that confuses you. Trying to understand something better it is how you grow as a chess player. I get what you are saying but I can't say I agree with it at all. It may not be the optimal time sensitive way for improvement but who cares.
The OP is more worried about opening the a-file than the doubled pawns it creates. I think that in itself is a great learning opportunity for them.
White is lost either way and the puzzle is made for black so white will make mistakes. Best play from both sides is probably this:
Also notice how Qb3+ avoids your line completely and white will still be able to castle regardless of what you do.
But it does seem that you take the weakness of doubled pawns too lightly.
I'd actually considered that option once Qb3+ I messed up the move after quite a few times.
That's completely possible. I know double pawns and generally bad because they can't defend each other and one blocks the other from advancing, but I don't know about other weaknesses to them. I tend to also ignore pawns unless they're impacting a move though.
Meh... there is definitely learning to be had by studying both sides of puzzles. It all depends on if YOU understand what is happening or not. There is nothing wrong with trying to understand something that confuses you.
Most certainly, good puzzles! But chess.com puzzles are not good puzzles in any sense professional puzzle makers (which we call problemists) would assign that term. When you read the OP's comment carefully, you'll see he is not confused by the puzzle at all - he understands it quite correctly. He is confused by what chess.com does with the puzzle solution and wonders why chess.com is doing that. That is precisely the question I answered. For reasons of its own (some related to teaching) chess.coms counter play is not always "the best" by human or engine standards. Studying a puzzle from both sides of the isle is good training but it will not remove the confusion on why chess.com played the solution in one way or another.
Meh... there is definitely learning to be had by studying both sides of puzzles. It all depends on if YOU understand what is happening or not. There is nothing wrong with trying to understand something that confuses you.
Most certainly, good puzzles! But chess.com puzzles are not good puzzles in any sense professional puzzle makers (which we call problemists) would assign that term. When you read the OP's comment carefully, you'll see he is not confused by the puzzle at all - he understands it quite correctly. He is confused by what chess.com does with the puzzle solution and wonders why chess.com is doing that. That is precisely the question I answered. For reasons of its own (some related to teaching) chess.coms counter play is not always "the best" by human or engine standards. Studying a puzzle from both sides of the isle is good training but it will not remove the confusion on why chess.com played the solution in one way or another.
Fair enough! Good points. I can't speak to whether the OP is confused or not though. It seemed that they had some inaccuracies in their analysis of the position but I could very much be wrong.
https://www.chess.com/puzzles/problem/1110500
https://www.chess.com/analysis?fen=r2q1bnr/pp1n1kpp/2pp4/8/3PP1b1/5N2/PP3PPP/RNBQK2R%20w%20KQ%20-%200%208&flip=true
You are Black. Solution is: Ng5+, Qxg5, Qb3+, Be6, Bxg5, Bxe6. The puzzle stops at Bxe6 but as white, axb3 seems the natural next move, which opens the a file for the rook.
Now back to the beginning...I was expecting and looking to play this line: Ng5+, Qxg5, Bxg5, Bxd1, Kxd1. There is no difference in material (black still loses the queen and bishop on the move after the puzzle, and white still loses the knight and queen), but now the a file rook is still pinned, and the king cannot castle if it takes on d1. If king doesn't take on d1, then you're up a bishop.