I like your question a lot. Though I don't play either side of any
1. e4 e5 openings, it got me thinking about the motivations behind playing the Scotch Game. Though you were clearly happy with the answer you accepted, I'm adding another answer because I don't personally understand how that one addresses the question as you asked it, and I think some further thoughts might be useful for others who come upon the question. So here's an alternative way of looking at things.
To answer your question, yes, against a peer, the Scotch Game really does offer chances of obtaining an opening advantage; Kasparov's record using it against his peers, including Karpov, is a case in point (even though, yes, he is Kasparov and we are not). The Scotch Game will not lead to an opening advantage against equal play by your opponent, but that is no different than with any opening. For example, that is just as true for the Scotch Gambit which Tony's answer recommends as an alternative. (And, incidentally, if the opponent declines the Scotch Gambit, you can just end up in the Italian Game anyway. If we're dismissing out of hand the "standard positional play" in the Scotch Game as offering no hope for an advantage, as expressed in Tony's answer, then I don't know why we wouldn't do the same for the Italian Game. No reason has been given for dismissing either.)
Regarding Kasparov's assertion that only the Ruy Lopez and the Scotch Game are "serious" attempts at obtaining an opening advantage for white after
1. e4 e5 2. Nf3 Nc6, it is safe to say that he was speaking in terms of what he considered to be a serious attempt where he lived, i.e. the very highest echelon of the chess world. I am of course completely unqualified to confirm or dispute Kasparov's opinion on that matter; judging from the frequency of which openings appear after those moves, it does seem that that is the consensus among top players.
In any case, while I won't compare or contrast the Scotch Game with those "lesser" alternatives, I will try to give you a bit of an answer to your question as to what kind of an advantage white can seek in the Scotch Game. As stated above, I don't typically play either side of this opening, and even if I did, I'm not a strong enough player to really be trusted. So I decided to look up John Emms' introductory book Starting Out: The Scotch Game; the thoughts below come from the beginning of that book, and all opinions expressed here are those of Emms, not myself. (If you find any of this explanation enlightening, then consider it a good advertisement for the rest of his book.)
3. d4 exd4, white has induced black to give up the center, and we have this pawn structure:
[FEN "4k3/pppp1ppp/8/8/4P3/8/PPP2PPP/4K3 w - - 0 1"]
This structure is already inherently better for white, with greater control of the center and more space for the pieces. Since black typically wants to avoid
4... Nxd4 5. Qxd4, which centralizes white's queen in a nice way, white often ends up exchanging the knights himself with
Nxc6, when black has two choices for recapturing. If
...dxc6 is played, then we have the following pawn structure, which is also typical of the Exchange Variation of the Ruy Lopez.
[FEN "4k3/ppp2ppp/2p5/8/4P3/8/PPP2PPP/4K3 w - - 0 1"]
Here white's structural superiority offers a significant advantage in the long-term. For instance, most king-and-pawn endings in this structure are wins for white, because of the ability to create a passer on the kingside, while black cannot do so on the queenside. Everything's not as simple as that, of course, as black can develop pretty painlessly and so has good short-term prospects; but white's structural advantage is the basis for his optimism.
If black instead recaptures via
...bxc6, then her central control is improved, and ideas of playing
...d5 are in the air. White often pushes with e5, and we have this structure:
[FEN "4k3/p1pp1ppp/2p5/4P3/8/8/PPP2PPP/4K3 w - - 0 1"]
Now white has even more space, and the e5 pawn cramps black's position. Black will try to free her position with a
...f6 pawn break, and white will try to prevent this, or make it as disadvantageous as possible.
In none of these scenarios does white obtain an advantage by force (though again, that's just chess for you), but we see that white has structural pluses which offer hopes for an advantage (especially in the long-term) in the Scotch Game. Here's a review of both Emms' book and a similar one by Gary Lane; if you want more guidance on this opening, that review, at least, is positive about both books.