Comparing 9...a6 to 9...b6 in the Panov-Botvinnik Attack This question is motivated by a recent LiChess game where I was black where I played the Caro-Kann and white played the Panov-Botvinnik Attack:

[fen ""]
[Site "https://lichess.org/J9cdDkL5"]
[Date "2019.03.26"]
[White "pierremarc1986"]
[Black "becky82"]

1. e4 c6 2. d4 d5 3. exd5 cxd5 4. c4 Nf6 5. Nc3 e6 6. Bg5 Be7 7. Nf3 O-O 8. Bd3 dxc4 9. Bxc4 b6


Here I played 9...b6 (with the idea of playing either 10...Bb7 or 10...Ba6 depending on how I evaluated the position after my opponent's move) and didn't even think to play 9...a6 during the game, which is both the prominent book move, and among Stockfish's top recommendations.

Question: Is 9...a6 better for black than 9...b6 in this line of the Panov-Botvinnik Attack?

The move 9...a6 prepares 10...b5, where black fianchettos the queenside bishop with more queenside space, but it costs a tempo.  It also allows 10. a4 which prevents 10...b5.

The main book lines on LiChess are as follows:

[fen ""]

1. e4 c6 2. d4 d5 3. exd5 cxd5 4. c4 Nf6 5. Nc3 e6 6. Bg5 Be7 7. Nf3 O-O 8. Bd3 dxc4 9. Bxc4 a6 10. a4 (10. O-O b5 11. Bb3 Bb7) (10. a3 b5 11. Ba2 Bb7) Nc6 11. O-O Bd7
This will be a short answer as unfortunately I don't have much time (last minutes of my break :) ).

In a nutshell: Both options are quite playable but result in slightly different structures, so it's up to you to decide which one leads closer to something you're more familiar with. As to why both moves 9...b6 and 9...a6 are equalizing for black, it lies in the fact that white has not castled yet and therefore they cannot really emphasize on black's slow development and there are no immediate pawn breaks left for them in the centre. All of which means, that extra tempo white has to invest for castling allows black this added flexiblity in choice. 

A slightly more concrete look at some lines: (please do check everything here with an engine, I don't have time/option to do that now)


White has no immediate ways of challenging b6: for example, with the king still in the centre, the immediate d5 (i), trying to liquidate the IQP and open up the centre bears no relevance as it easily leads to a much better endgame for black (with all the initiative):


(i)

 [title "immediate d5 after 9...b6"]
 [fen "rnbq1rk1/p3bppp/1p2pn2/3P2B1/2B5/2N2N2/PP3PPP/R2QK2R b KQ - 0 10"]

 1...Nxd5 2.Bxd5 exd5 3.Qxd5 Qxd5 4.Nxd5 Bxg5 5.Nxg5 Re8+ 6.Ne3 Ba6 {forcing white to castle long} 7.O-O-O Nc6 8.Kb1 Nb4 {thanks to weakened light squares black has great initiative, trade 1 pair of rooks and plans of Bd3-Bg6 following by Nd3 are always out there, where white's minor pieces can only be defensive.}



Now let's see what difference it would make if in the same position as (i) white's king was already safe/castled:


(ii)

 [title "Same position as (i) but assuming white's castled short already"]
 [fen "rnbq1rk1/pp2bppp/4pn2/6B1/2BP4/2N2N2/PP3PPP/R2Q1RK1 b - - 0 1"]

 1... b6 2. d5 Nxd5 3. Bxd5 exd5 4. Bxe7 Qxe7 5. Nxd5 Qd8 6. Rc1 Na6 7. Ne5 {A completely different end result, both sides have quite a healthy structure but if anyone it's white with an extra positional edge, in contrast to previous case where black was leading a better endgame.}


These lines underline the fact that white cannot really commit to much in the centre here (in order to pressure black by any means) without castling first, therefore black really does have an extra tempo at their disposal to choose either the immediate b6 or the somewhat slower but more sensible a6 (as it also provides important control over b5, which can still be followed by b6-Bb7 with the added advantage that you're in time to respond to a4-a5 with b5).


A tad more on the 9...a6 line followed by a simple developing plan leads to a very decent position with active ideas for both sides: white's play should revolve around emphasizing the IQP to establish a centralised knight on e5 (in other words making use of the extra dark square control provided by d4), and in contrast black's play is to exploit the IQP structure in order to establish a minor piece on d5 blockading the pawn and keeping the prospect of the d4 pawn potentially becoming a weakness, while activating the somewhat passive d7 bishop. 


(iii)

 [title "One possible continuation after 9...a6 Nc6 Bd7"]
 [fen "rnbq1rk1/pp2bppp/4pn2/6B1/2BP4/2N2N2/PP3PPP/R2QK2R b KQ - 0 9"]

 1...a6 2. a4 Nc6 3. O-O Bd7 4. Qe2 Nb4 {preparing Bc6 to d5, one way or the other black is here in time to establish a piece on d5 blockading d4's advance} 5.Ne5 Bc6 6.Rad1 Bd5 7.Nxd5 (7.Bxf6 Bxf6 8.Nxd5 Nxd5) Nfxd5


Finally, not much else to say about b6 as it is equally decent: the light square fianchetto is quite fitting for black in all these kinds of IQP structures, as white often cannot easily challenge the bishop by either trading it or closing the diagonal, and on the other hand black ends up with a well active bishop which also provides additional control over d5.