For me, spider bites don't itch so much as they hurt. I think I'm allergic to them, though. Every time I've had one, it's always swollen up and been hot enough that I could melt a couple ice cubes on them before it would slow down on the rate of melting. The bite marks looks different from a mosquito, considering there's two very close together marks instead of one like a mosquito bite would have. So long as you don't have any symptoms like the above poster mentioned, however, you shouldn't have anything to worry about.
I've gotten tons of spider bites before. For the most part they're similar to a mosquito bite, raised, irritated and a little hard. However, in my opinion they itch less and I do have less of a reaction to them. The mass majority of spider bites are nonvenomous so unless you start feeling extreme burning, pain, nausea or become feverish there's no need to worry.