In the Berlin Defence, why is 4.0-0 preferred to 4.Nc3, as then White can transition into the four knights?
As Phonon said in his comment:
“The transposition from the Berlin Defense (1.e4 e5 2.Nf3 Nc6 3.Bb5 Nf6 4...) to the Four Knights Game (Spanish variation 4.Nc3) is a perfectly valid one, but the four knights game is known to be a quite dry opening for white, as in there are various easily equalizing lines for black, starting from the Rubinstein 4...Nd4 (see Kramnik games). Instead 4.O-O is the more critical line (one that offers more hanging fruits...) alongside the 4.d3 line which is known as an anti-Berlin approach.”
As already said in some answers, 4. Nc3 is relatively harmless as it transposes into the four Knights game. It is a shame because developing the Knight on c3 might seem very logical (it follows the generic principles of the opening !).
Instead, 4. O-O is forcing Black to play a committal move, as 4... Be7 leaves Black with an unpleasant position:
[FEN "r1bqkb1r/pppp1ppp/2n2n2/1B2p3/4P3/5N2/PPPP1PPP/RNBQ1RK1 b kq - 5 4"] 4... Be7 5. Nc3 d6 6. d4
And White is doing more than fine here.
So Black doesn't really have any other choice than burning bridges in some way (actually, Black is doing fine, of course) and go for 4... Nxe4 that changes the pawn structure and makes the game a lot less symmetrical, going into the Berlin endgame in which White can at least press.
[FEN "r1bqkb1r/pppp1ppp/2n2n2/1B2p3/4P3/5N2/PPPP1PPP/RNBQ1RK1 b kq - 5 4"] 4... Nxe4 5. d4 Nd6 6. Bxc6 dxc6 7. dxe5 Nf5 8. Qxd8 Kxd8
With just too much theory here, but that's another story !
To know more about how to press with White on the Berlin defense, my Berlin defense memo explores many solutions, including 5. d4 but also 5. Re1, and one move earlier, the possibility 4. d3:
[FEN "r1bqkb1r/pppp1ppp/2n2n2/1B2p3/4P3/5N2/PPPP1PPP/RNBQK2R w KQkq - 4 4"] 4. d3
Such a modest-looking move, yet the other real possibility to play for a win with White !