Thymine has a methyl group which is not found in uracil. Having a methyl group means it is more stable. That is why thymine is more stable that uracil.
http://www.mun.ca/biology/scarr/T_vs_U.gif
DNA is more stable than RNA because DNA has deoxyribose sugar and RNA has ribose sugar. So DNA has thymine base and RNA has uracil. Also, all genetic info is stored in DNA, so DNA has to be stable and 'long-lived' to stored the information. When you need a protein, you need this genetic info but DNA don't move out from nucleus, so you need RNA because after you get your protein made, you don't need this info anymore and RNA is degraded.