As title above, what is the point of playing the 5.d3 (or 6.d3) Ruy Lopez, when White can play Giuoco Piano (or Modern Italian)? In other words, why play 1.e4 e5 2.Nf3 Nc6 3.Bb5 a6 4.Ba4 Nf6 5.d3 (or 5.0-0 Be7 6.d3), when White can reach a very similar Pawn - and Piece - structure in Giuoco Piano (or Modern Italian), or in the 4.d3 Two Knights, avoiding some uncommon (but potentially dangerous, and in any case demanding in terms of opening preparation) replies to the Ruy Lopez, such as the Schliemann Gambit, the Bird Defense, or - in the 6.d3 case - the Open Spanish (5-0-0 Nxe4), and others?
Giuoco Piano gives White slightly fewer chances for the initiative that White is supposed to have. Also the Two Knights defense almost completely equalizes the game at high level with simplifications: 1.e4 e5 2.Nf3 Nc6 3.Bc4 Nf6 4.d3 Be7 5.0-0 d6...(5.0-0 Nxe4 6.Re1 d5 7.Bxd5 Qxd5 8 Nc3 Qa5 9.Nxe4 Be6 10.Neg5 0-0-0 11.Nxe6 fxe6 12.Rxe6...) After 5.0-0 Black can also choose to play complicated and crazy Max Lange Attack. White may not like it, to put it mildly. The Ruy Lopez gives more options and more chances for White. And the Open Ruy Lopez 1.e4 e5 2. Nf3 Nc6 3.Bb5 a6 4.Ba4 Nf6 5.0-0 in case of 5... Nxe4, is fine for White: 6.d4 b5 7.Bb3 d5 8.dxe5 Be6 - White maintains some initiative. The same holds for the Bird Defense.
As to the Schliemann Gambit, it doesn't have to be dangerous: 1. e4 e5 2. Nf3 Nc6 3. Bb5 f5 and after 4. d3 the gambit fizzles out. It's a typical problem with many gambits: they often can be neutralized. It's also one of the main reasons that they fell out of fashion at high level. The idea is not to hold on to the sacrificed material but return it and equalize the position, so that White gets nothing, no initiative out of the opening. Now then with modern engines we know it's usually better to hold on to the material and play precisely but it's hard and players might be a bit rusty with gambits. It's easier for humans to avoid risks and complications. A simple example is the Danish Gambit 1.e4 e5 2.d4 exd4 3.c3... and now Black can nip it in the bud with 3...d5 leaving White with close to 0.00. The King's Gambit is more sound as is the Giuco Piano but it's still fewer winning chances for White at high level in comparison with well-reputed solid openings such as the Queen's Gambit, Ruy Lopez, Sicilians, etc.
However, it doesn't mean that gambits are no good and folks should avoid them. It's just that well-reputed openings are preferred at high level. Again, at club level, various gambits and tricky openings are fine, and they are fun to play, such as the King's Gambit, Giuoco Piano, Smith–Morra Gambit, Albin Countergambit, etc. Yet, gambits are rare at high level. They are more of a surprise value, not the main repertoire.
Sometimes rush, tricky play, typical for a club player is castigated. For example, if I'm not mistaken Tarrasch said something, like, after 1.e4 e5 2.Nf3 Nc6 3.Bc4 Nf6 the move 4.Ng5 is a move of a fool. That seems too harsh but his point is that less skilled players often rely on tricky lines or launch headlong attacks prematurely. This often works at low level but backfires against professionals. Even at serious club level such trickery is often stalled.
Disclaimer. I'm not against gambits as this answer may suggest. I like gambits and thrilling games, and I think amateurs can benefit from them and have more fun playing chess.