The difference between Halo 3 and ODST is that Halo 3 continues the series. ODST gives you a side story by putting you in the boots of a different character in the series, mainly a squad of Helljumpers (one at a time of course). The UI is slightly different, and gameplay mechanics are similar but if you're a fan of mystery/film noiresqe stories, you'll enjoy ODST. Also, ODST was the first in the Halo series to include firefight, so you'll be able to enjoy the evolution of the mode throughout the series.
[quote name='dirtyvu']Totally forgot about ODST. I didn't like that game. Story was slow. Too many levels could be skipped by rushing through without killing anything.[/QUOTE]
You have to keep in mind that there was a different focus of the game. Because you play as a "normal human", you have to have a different mindset and playstyle going into the game. You're not going to be rushing headon into battle due to the main characters not being super soldiers; you get to pick and choose your battles, and therefore have that ability to simply skip/avoid some of the battles in the game. That being said, you can also choose to fight and kill everything as you would if you WERE the Master Chief, but once again you have to use a different set of tactics because you're slightly more vulnerable than the Chief is.
As for the order of playing the games, there's two main orders you can play: story order, and release order.
Story order: Halo Reach, Halo 2, Halo ODST, then Halo 3.
Release order: Halo 2, Halo 3, Halo ODST, Halo Reach.
Personally, I'd suggest a mixture of the two: Halo 2, Halo ODST, Halo 3, Then Halo Reach. Reach last because there's several references that are made that will make more sense if you've played the rest first. Also while you're at it, if you're really invested in the story and like to read, I'd pick up Halo: The Fall of Reach, First Strike, and Ghosts of Onyx. Fall of Reach gives you more insight as to the Master Chief's background and what happened to the rest of Reach, First Strike bridges the events between Halo 1 and 2, and Ghosts of Onyx semi-bridges Halo 3 and Halo 4, the rest of that story is slowly being released as a series in the next few years. If you don't read them, it's okay because the games tell the story really well, but reading the books give you better insight as to why things happen the way they do.
If you want even more backstory, another trilogy is being written which I highly endorse, as it's well written and talks about the forerunners. Cryptum and Primordium are the first two in the series. The last is being released sometime this year.
TL;DR ODST mystery style Halo game, with a different main character and starts the firefight mode. Suggested order: Halo 2, ODST, Halo 3, Reach. You should read the books as well, preferably in order (Fall of Reach and First Strike before Halo 2, Ghosts of Onyx after Halo 3/before playing Halo 4).