This is merely too long to be a comment.

I'm not sure that I completely agree with or understand the basis of the question. Is this one of those questions where the OP is taking for granted that varieties are interesting and then wondering what schemes are useful for in this context? If so, there have certainly been other questions of this sort on mathoverflow. However, I get the feeling this is not what's being asked for, in which case I'm not sure that the example of character theory being used to solve a problem of group theory is completely analogous. In that example, groups are already inherently part of character theory, so it seems reasonable that problems of group theory might be attacked using character theory. When asking a similar question about schemes, I'm not sure what the starting point is. To me, asking what you can do with schemes is more like asking 'what can you do with groups?' than 'what can you do with characters?'. At this point, I would say that both groups and schemes are naturally occurring basic mathematical objects that are of great interest to many mathematicians.