For difficult problems you can often write a brute force solution with terrible complexity (think 2^n) for the small input set. For the large though, you'll need to really understand the problem and write an efficient solution via memoization, dynamic programming, pruning, or good (sometimes obscure) algorithm choices. For small solutions you get instant checking of your solution on submit (but not what you failed on, only that you did), for large ones you submit and hope for the best.

As for why you should care, for me at least it's fun to come up with fast solutions to problems regardless of whether they're efficient or if my code is well designed. You learn about new types of algorithms and ways of solving problems. For example, dynamic programming was foreign to me until I started doing tons of TopCoder. Google also uses it for recruiting, I'm still getting contacted by them even though I skipped last year.

My favorite moments come from solving problems poorly while knowing there's a proper solution out there. Ex: There was a problem in some competition where it was some special way you had to return the y-intercept of a line with a given slope that bisected a particular polygon. There was an actual way to do it properly, but I'm not so great at math/geometry so I ended up doing a binary search and did enough iterations to get the right amount of floating point precision they wanted.