Does depth (number of ply) still matter once it is high enough? This isn't real, just imagine this happening.

Computer A is supercomputer. It can calculate 30 ply deep within 20 seconds.

Computer B is supercomputer. It can calculate 15 ply deep within 20 seconds.

They play against each other chess.

Does these 15 depths really matter? I guess that within these 15 depths there may be trillions way to escape a checkmate or capture of important piece. Sure, Computer A knows more. But Computer B is able to predict future quite far too, in my opinion, far enough to defend himself really well.
Yes, those 15 depths very much matter. 

Consider this position that occurred in Kasparov's Immortal Game vs Topalov. 

  [White "Kasparov"]
  [Black "Topalov"]
  [FEN "b2r3r/k4p1p/p2q1np1/NppP4/3R1Q2/P4PPB/1PP4P/1K2R3 b - - 0 24"]


I tested this position with several engines. Some engines, at depth 15, failed to detect that 24...cxd4 is a losing move and thought it was winning. Those same engines, at a greater depth, played the correct move 24...Kb6!

For example, even an engine as powerful as Stockfish 4 initially at depth 21 thinks the losing move 24...cxd4 is correct. 

Stockfish DD 64 SSE4.2: 24...cxd4 25. Re7+ Kb6 26. Qxd4+ Kxa5 27. Qc3+ Kb6 
28. Qd4+ Qc5 29. Qxf6+ Bc6 30.Qxc6+ Qxc6 31. dxc6 Rd1+ 32. Ka2 f5 33. c7 Rc8 
34. Rxh7 Rxc7 35. Rh6 Rc6 36. g4 f4 37. g5 Rd2 38. c3 Rxc3 39. Rxg6+ Kc5 
40. Bg4 Rcc2 41. Rxa6 Rxb2+ 42. Ka1 Rbc2 43. Kb1 
(-1.45/21)


The same engine, when kept on for a little more depth, shows 24...Kb6 to the correct move. 

Stockfish DD 64 SSE4.2: 24...Kb6 25. b4 Qxf4 26. Rxf4 Nxd5 27. Rxf7 cxb4 
28. axb4 Rhe8 29. Rxe8 Rxe8 30. Nb3 Re1+ 31. Kb2 Re2 32. Rxh7 Nxb4 
33. Kc3 Nd5+ 34. Kd3 Rxh2 35. Rh4 Ne7 36. Nd4 Nc6 37. Nxc6 Bxc6 38. f4 Kc5 
39. Be6 Rxh4 40. gxh4 Bd5 41. f5 gxf5 42. Bxf5 a5
(-0.78/26)


Fritz 11 SE, at depth 15, also failed. But it found the correct move at depth 16! 

Fritz 11 SE: 24... cxd4 25. Qxd4+ Qb6 26. Re7+ Nd7 27. Qe5 f6 28. Qc3 Qg1+ 
29. Ka2 Bxd5+ 30. Nb3 f5 31. Qc7+ Ka8 32. Rxd7 Rxd7 33. Qxd7 Bxf3 34. Qd6 Qa7  
(-1.44/15) 

Fritz 11 SE: 24... Kb6 25. b4 Qxf4 26. Rxf4 Nxd5 27. Rxf7 cxb4 28. axb4 Nxb4 
29. Nb3 Bd5 30. Rf6+ Nc6 31. Nd4 Rdf8 32. Rd6 Kc5 33. Rxc6+ Bxc6 34. Ne6+ Kd6 
35. Nxf8 
(-0.59/16)


Also consider this incredible problem like position that I found here. 

Stockfish was not able to find the winning line 1. Be2+! until depth 31 and up until then it thought it was a bad move. I demonstrate the win here. The point is that Black is in zugswang because of mate threats and has to either give up the queen or move the a pawn which would allow White to create a passed pawn and win. 

   [FEN "8/2p1q3/p3P3/2P4p/1PBP2kP/2N3P1/7K/8 w - - 0 1"]

   1. Be2+ $1 Kf5 2. Nd5 $1 Qxe6 3. Bd3+ Kg4 4. Be4 $3 Qh6 (4... Qxe4 5. Nf6+) 
   5. Nf4 Qg7 6. Nd3 $1 Qxd4 7. c6 $1 a5 (7... Qxe4 8. Nf2+ Kf3 9. Nxe4 Kxe4 
   10. Kg2 Kd4 11. g4 hxg4 12. h5 Ke5 13. h6 Kf6 14. Kg3 Kg6 15. Kxg4 Kxh6 
   16. Kf5 Kg7 17. Ke6 Kf8 18. Kd7 Kf7 19. Kxc7) 8. b5 $1 a4 9. b6 cxb6 
   10. c7 Qxe4 (10... Qc3 11. Nf2#) 11. Nf2+ Kf3 12. Nxe4 1-0


Here's the engine log from Stockfish 4. As you can see, it detects that 1. Be2+ is winning, only on depth 31! 

Stockfish DD 64 SSE4.2: 1. Be2+ Kf5 2.Bc4 c6 3. Ne2 Qf6 4. Kg2 Kg4 5. Nf4 Qxd4 
6. Bd3 Qe3 7. Be2+ Kf5 8. Bf3 Qd2+ 9. Kh3 Qxb4 10. e7 Qe1 11. Ne2 Qf1+ 12. Kh2 Qf2+
13. Kh3 Qe3 (-1.05/22) 

Stockfish DD 64 SSE4.2: 1. Be2+ Kf5 2. Bc4 Qf6 3. Ne2 c6 4. Bxa6 Qxe6 5. Bd3+ Kf6 
6. Nf4 Qe1 7. d5 Qxb4 8. dxc6 Qxc5 9. Be4 Ke7 10. c7 Kd7 11. Nd5 Kd6 12. Kh3 Qc4 
13. Bg2 Qg4+ 14. Kh2 Qc8 15. Be4 (-1.15/26) 

Stockfish DD 64 SSE4.2: 1. Be2+ Kf5 2. Bc4 Qf6 3. Ne2 c6 4. d5 cxd5 5. Bxd5 Qb2 
6. Bc4 Kf6 (-1.01/28) 

Stockfish DD 64 SSE4.2:  1. Be2+ Kf5 2. Nd5 Qxe6 3. Bd3+ Kg4 4. Be4 Qh6 5. Nf4 Qf6 
6. Nd3 Qxd4 7. c6 Qxe4 8. Nf2+ Kf3 9. Nxe4 Kxe4 10. Kg2 Ke5 11. Kf3 Kf5 12. g4+ Kf6
13. gxh5 Kg7 14. Kf4 Kf6 15. h6 Kg6 16. h5+ Kh7 17. Kg5 Kg8 18. h7+ Kxh7 19. Kf5 Kg7
20. Ke6 Kh6 21. Kd7 Kxh5 22. Kxc7 Kg5 23. Kd7 (6.06/31)