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Bc4 vs Bb5

Just wondering after 1.e4 e5 2.Nf3 Nc6 3.Bb5 a6 4.Ba4 b5 5.Bb3 doesn't this look like a better version for black than if white had played 3.Bc4 instead?
Yes, but black usually plays 4...Nf6, while 3 Bc4 Nf6 allows 4 Ng5.
There are good lines against 3. Bc4. 3. Bb5 is more aggressive and leads to more lasting pressure.
So if you played Bc4 you'd be happy if black could then play a6 and b5 in one move?
Robin22, I would feel neutral as this line is average for white, just as a Giuoco Piano would be. The strange point is that the Two Knights Defense is considered better for black than Ruy Lopez, the Max Lange, and the Giuoco Piano.
Yeah I'm just asking why since it looks like the same position except black has played a6 and b5 and white's bishop is on b3 instead of c4. Shouldn't blacks position be better with a6 and b5 and therefore worse for white than if he/she played 3.Bc4 instead?
I know I'm wrong I'm just curious
Having the bishop on b3 (or b5 or a4) instead of c4 often allows white to leave the e4 pawn "hanging" after black plays Nf6. The idea is to answer Nxe4 with d4, opening the position with black's king still being in the center.
With a bishop on c4, this idea usually doesn't work because black could play d5 with tempo after taking on e4.

More information:
youtube . com / watch?v=2hriSNXPkIA

Edit: Let me add that having played a6 & b5 doesn't neccessarily help black. It does give him more space on the q-side, true, but it also gives white an additional pawn break with a4.
"the Two Knights Defense is considered better for black than Ruy Lopez, the Max Lange, and the Giuoco Piano"
This seems wrong: most top grandmasters play 3...Bc5 instead of 3...Nf6 after 3 Bc4, so they avoid the Two Knights Defence and aim for the Giuoco Piano.
Back to the Original question: after 1 e4 e5 2 Nf3 Nc6 3 Bb5 a6 4 Ba4 b5 5 Bb3, what should black play?
5...Nf6 6 o-o and now what?
5...Bc5 6 Nxe5 Nxe5 7 d4
Also in the giuoco piano white voluntarily plays Bc4-b3. So it is doubtful if ...a6 and ...b5 are more useful than Bb3.
To expand on Dr_King_Schultz, the bishop is safer on b3 than on c4. The attack by playing d5 is the easier example, and the attack by Na5 is rare but also effective. By having the bishop on b3 also allows for an easier transfer to the b1-h7 diagonal. By avoiding the center fork trick, it becomes easier to play d4 without wasting a tempo on d3.

In all variations of the Ng5 variation of the Two Knights, except for the Lolli and Fried Liver Attack, black gets the imitative and a better game for the cost of a pawn. In all variations of the Ruy Lopez, black is struggling for a draw.

BTW, the original question is to explain the difference between Bc4 and Bb5, it has nothing to do with what black should play in the Two Knights Defense.

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