The biggest difference is in the story line. In Halo 3, as in Halo and Halo 2, the story is about the Master Chief, a Spartan, who is somewhat of a cyborg. His armor is powerful and has quite a few special features. You get the feeling that his suit is part of him.
ODST is about Orbital Drop Ship Troopers. They have suits similar to the Master Chief, but with some differences. As far as I have played it, it all takes place on Earth, where Halo, Halo 2, and Halo 3 take place mostly on "Halos" or objects similar to Larry Niven's Ringworld.
The story line is basically that humans are at war with the Covenant. The Covenant are a race (?) that are guided by a religion. Some members of the religion want to fulfill an ancient prophecy by activating the Halos. However, these artifacts are actually designed to kill off all life in the galaxy. That is because there is a sort of parasitic entity known as The Flood, that takes over sentient creatures, and makes them into zombies, but fast and capable zombies. Eventually, in Halo 2, and more so in Halo 3, some members of both the human and Covenant races work together to fight the Horde and stop the activation of the Halos.
I don't know where in the timeline ODST actually lies, but so far all of the combat that I have experienced is on earth against Covenant elements. You play a Drop Ship Trooper, dropped into a large city, but separated from all of your platoon. You are a Rookie, and have to solve various puzzles and problems as you try to reunite with other Marine elements.
The real attraction of all of the Halo games is in the multiplayer capabilities. Halo provided great local multiplayer scenarios, and there was even a way to connect multiple XBox consoles. Halo 2 improved on it, and provided limited connectivity through the internet. However, Halo 3 really shines with its online connectivity and playability.
ODST has the same level of online play, and offers a lot of new player areas and scenarios. However, it also has a "Firefight" scenario where multiple players are helping each other to survive waves of Covenant. All of the other games focus on player vs. Player scenarios such as Deathmatch, Team Deathmatch, and Capture the Flag.
I have not played either Halo 3 or ODST in the multiplayer mode. When I watch my boys play, I don't really like it much, as I am a slow and careful player in first person shooters. I like to take my time and set up in spots where I can snipe or catch others as they rush around. Because of this, I used to always beat them when playing the earlier games. Halo 3, however, is hectic, and appears to require that the characters are always on the move. There don't seem to be any places where a player can "camp" and not have someone come up behind them.
I really like first person shooters, and the Halo series are about the best out there. Halo 3 and ODST are the latest generation of the best.