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Which one is the better drawing weapon?
Which one is harder to beat?
Which one is more solid?
Which opening should black pick if he/she doesn't want to lose?
If black needs a draw in the last round to win the tournament, which opening should black pick?
Before I address the questions, it's important to understand that the Berlin and Petroff are different openings in the sense that one cannot force the Berlin to appear over the board in a King Pawn opening. The Berlin is a variation of the Ruy Lopez, whereas the Petroff deviates from any Spanish variations on move two with 2...Nf6.
Historically, the Petroff deals consistent results at the club, tournament, and world-class levels. However, it's known to be a relatively unambitious opening with a solid reputation (until recently, where more aggressive lines have been discovered for White, namely variations after the move 5. Nc3).
The Berlin Defense is also known for its solidity, but it is dependent on White playing 3. Bb5. Kramnik used it as a drawing weapon against Kasparov at the 2000 Classical World Chess Championship.
However, I do not want to make a recommendation purely based on the nature of the two openings. Investigate the two openings yourself and play them in a few online games to get the feel of the resulting positions. If you find yourself comfortable more with one over the other, then you should pursue that opening. As an aside, I wouldn't "play to draw" unless there's significant money on the line or a national title - you'll learn more by playing good chess without an end result in mind.
There no certain answer to your question. It depends.
An opening with a "drawish reputation" usually turns against the player who chooses it without having a deep knowledge of the opening itself.
That said, I think that both of these, if understood, can be a very hard match against a White player who desperately tries to win. But they're not easy to play.
In my opinion, as a 1. e4 player (so, with the "opponent's point of view"), the Berlin is quite solid but harder to play: Black has to find suitable squares for his pieces, and must play carefully to avoid White attacks.
As White, I am comfortable playing against the Petroff: for Black it's a little bit easier to play, but has many traps, and usually the Knight becomes an easy target for White pieces. Black will lose tempo while trying to defend it.
In conclusion, don't choose an opening only for its "drawish" reputation: this may be true only at higher levels, where players know very well almost every opening. At lower levels, on the contrary, the player who relies on a specific "drawish opening" in order to draw a game will find himself in trouble more often than not.
If black needs a draw in the last round to win the tournament, which opening should black pick?
If a player only needs a draw in the last round, he or she should play normally (of course this is easier said than done) and only sometimes remember that to reach the goal they only need a draw, for example if they can force draw by perpetual. This means the player should play the opening they are most comfortable with. If the only opening black knows is Sicilian Najdorf, then they should play that.
Moreover, I don't think there is really a drawing weapon for under 2000 players; I believe draws a fairly rare at that level. Having said that, I think that the Berlin is more "drawish" in the since that it leads to exchange of queens (a least in the main-line). I don't think that the Petrov is really all that drawish, usually the position is pretty open with a lot of piece activity and opportunities for tactics.
The reason the Petrov has a drawing reputation is that many top-level GMs use it to draw. However, the main reason it works for them is that they know the Petrov really well, and often can show majic with tactical nuances, not because the position is really drawish or super solid. Even Kramnik had some quick losses with the Petrov (see Anand-Kramnik below), when the opponents had out-prepared him, those positions are tricky.
The Marshall gambit in Ruy Lopez is another opening top-level GMs often use to draw; however, that opening is hardly drawish or solid for sub 2000 players.
Here is is Anand-Kramnik game mentioned above:
[FEN ""]
[Event "Mtel Masters"]
[Site "Sofia BUL"]
[Date "2005.05.19"]
[EventDate "2005.05.12"]
[Round "7"]
[Result "1-0"]
[White "Viswanathan Anand"]
[Black "Vladimir Kramnik"]
[ECO "C42"]
[WhiteElo "2785"]
[BlackElo "2753"]
[PlyCount "39"]
1. e4 e5 2. Nf3 Nf6 3. Nxe5 d6 4. Nf3 Nxe4 5. d4 d5 6. Bd3 Nc6
7. O-O Be7 8. c4 Nb4 9. Be2 O-O 10. Nc3 Bf5 11. a3 Nxc3
12. bxc3 Nc6 13. Re1 Re8 14. cxd5 Qxd5 15. Bf4 Rac8 16. Qc1
Na5 17. c4 Qe4 18. Bd1 Qd3 19. Re3 Qxc4 20. Re5 1-0
