Two things, really: Blu-Ray holds a lot more data, and they have a better protective coating, so resist damage better.
Because they hold a lot more data, they can fit HD versions of movies on the Blu-Ray discs, taking full advantage of the high definition capabilities of modern HD TV's. If you have a HD TV, in most cases -- unless the studio was cheap and just dumped the DVD data to Blu-Ray, which I haven't heard of yet -- you should see a big difference in picture quality from DVD to Blu-Ray.
If you have a standard old TV -- not a HD TV -- then you may not see much of a difference, other than additional extras often put on Blu-Ray (again, taking advantage of the extra space). Old TV's just are pretty low resolution, so you can't see much of the extra detail.