Personally, I use Pandora more a month than Last.FM ever, but the former is only available in the US. Last.FM offers tag searching for more general listening, when an artist or song isn't a concern, but more a genre or class of music. Since they're based in Britain, however, Last.FM can't offer pausing of music, only stopping, which is a major turnoff every time I fire up the site or app. For the same reason, however, I also assume that other music services won't follow the same model.
Pandora has, for a while now, had ads in the free version and a hours/month restriction, but with AdFree installed and even with daily listening I haven't felt the need to pay a dime for it. Spotify is another cool service that lets you listen to millions of songs and even download them, for free (at least last time I checked). It's less geared towards music discovery (I've found at least 50 bands and artists I would not have heard of on Pandora by listening to music similar to theirs), but it's great if you know what you want to listen to and want it then and there.
I don't know how much Last.FMs move affects you, but I can live with it. I am curious though what your favourite music streaming service is and why, so if you would, post it in the comments.