HDMI supports HD resolutions, which you can do over component, but it's digital so you'll see less signal degradation if you're running long cables (to a projector at the opposite end of your room, say). HDMI also supports a few things that component doesn't, like HDCP copy protection, carrying audio along the same cable, and carrying LPCM audio so you can use the new HD audio formats on BluRay/HDDVD players.
If you want to watch HDCP copy protected content (an HD film from a cable box, say) you may find it won't actually display if you aren't using HDMI. Also, many TVs support higher resolutions over HDMI than they do via component - you may find your TV will do 1080p over HDMI but only goes up to 1080i over component, or whatever.
Beware when looking at specs though, not all HDMI connections are created equal - just because an amp has an HDMI socket, it doesn't necessarily mean it supports all the features that are possible over an HDMI connection. Most cheaper HDMI enabled amps won't do audio over HDMI or if they do they don't support LPCM, or they support it with a limited number of channels (my amp for example, the Onkyo 604E, supports up to 7.1, but it'll only do 5.1 via LPCM over HDMI).