What difference would I see with a more expensive bike?

Right now I have a Schwinn sidewinder that I got a walmart last year for around $150. I rode it alot and for $150 its a decent bike. But I would like to get into more serious bike riding this summer with a little rougher terrain and steeper climbs. I have about a $500 budget, and right now I'm looking into a Trek, possibly a 3700 Disc model and was just wondering what the main differences in performance would be between the $150 bike I have now and a bike in the $500 range? The main thing I am looking for is being able to climb steeper hills without much more effort, but don't know if this would be the case since both the bike I have now, and the models I am looking at are 21 speed. Would a better model 21 speed be better at climbing than the $150 21 speed I have now or would they be about the same since they are both 21 speed?