Sesshomaru hates Inuyasha because he bares the Tetsusaiga. Sesshomaru wants the sword for himself. He hates the fact that his younger brother is the rightful owner of the better sword [out of Tetsusaiga and Tenseiga], and that pisses him off. He also enjoyes taunting Inuyasha.
The two brothers are always fighting to test each others' strength. I believe Sesshomaru is secretly training his younger brother to become stronger. I mean, everytime they meet, Sesshomaru always asks for a fight, or knows Inuyasha will insist. Awwe. Now that's brotherly love.