Well, their first real exposure to Spartans resulted in the deaths of two ODSTs. Rumors probably got out, and now ODSTs are wary/hateful of Spartans.Yeah, but Chief didn't even know his own strength.The freakin initial invasion of earth included 2 CAS-class battlecruisers (About the size of the Infinity), and Chief pretty much singlehandedly blew one to bits with the covenant lightmass bomb. He destroyed Installation 04 by himself. He destroyed Installation 07's replacement and damaged the ark with the help of Arby and Johnson. He got help getting into the Didacts ship, but of course, he charged in head first and blew it up with a nuke. This guy deserves the Rank of Master Legendary Warrior of the Galaxy to the 7th power extreme.
Uh dude, he was on Installation 00 to destroy Installation 04B. 07 already existed (7 Halo Rings anyone? No?).
And OP, the reason why ODST's hate Spartan guts is because of the strained tensions. ODST's were indeed, the best of the best. Halo 3 ODST? "But we are the best of the best." Except the Spartans. Who are better. It's a motto. And its one thats hard to remove. Sure, they're grateful, some of them. But the rest are just like "Really? We had this here! D-bag." Ya get my point. The ODST's were like the Grade 12's of High School, then suddenly became a freshman for College. It's... jarring, to say the least.