What advantages do "alt-coins" have over bitcoin?
It depends a lot on the specific altcoin you're talking about. For the most part, there aren't a lot of major differences between Bitcoin and the alts. Sadly many of them exist only as the leftovers of pump-and-dump schemes intended to make their creators rich, though there are a few that are interesting in important ways.

One of the primary benefits of a thriving altcoin economy is that they often represent the testing of a theory that may eventually benefit Bitcoin itself. Many altcoins, for example, use scrypt as their mining algorithm instead of SHA256. If SHA256 is discovered to be compromised some day, having a drop-in replacement for that subsystem available would be a major boon.

You specifically mention faster confirmation times as a potential benefit, but the most often overlooked part of that statement is that transaction finality is a result of computational cost, not of time. This means that if an altcoin has one tenth the confirmation time and an equal amount of hashpower relative to Bitcoin it will take ten times as many confirmations to reach the same level of transaction security. Faster confirmations are a psychological benefit, not a technical one.

The "silver to Bitcoin's gold" theory, largely pushed by Litecoin proponents, is also fallacious in that the entire reason gold requires a baser partner metal is that it is not adequately divisible. Paying for a candy bar with gold would be incredibly difficult for both parties involved since the amount of gold needed would be absurdly small. Bitcoin is not subject to such physical limitations and as such can be divided into such tiny portions relatively easily.

Taking all of the above into account, it should be clear that while a handful noteworthy exceptions do exist, the majority of altcoins are intellectual curiosities at best and scams at worst. A few make meaningful changes, but thus far none of those changes has really been a meaningfully large improvement over the base system.