The same as the difference between a tissue and a kleenex, one is a brand name, one is a type of product.
AFAIK all i3's are dual cores with hyper threading, which gives them two physical cores, and 4 logical cores, hyper threaded cores arent as good as real cores, they are not quad cores.
Which i3 and which quad core processor are you looking at? Are you looking at Core 2 Quads?