You asked if there are any differences between antenna types and the answer is YES!
While you must have a dome if you want in-motion, all of the domes come with some kind of limitation compared to an open faced dish like the Trav'ler. While the Trav'ler is quite a bit more expensive, you cannot receive ANY DirecTV HDTV programming and if you are a Dish customer you will be limited to using one signal tuner receiver if you want to be able to watch what you want when you want.
Using a DVR or 2 separate receivers can be dome but you will not have full use since Dish uses 3 satellites and the domes only contain 1 LNB. Let me know if you need additional details about this.
If you connect a DVR to your Trav'ler you can watch record programming while traveling it you are just looking for a distraction for your passengers.
The domes suffer the most during any sort of weather event and that tends to be when I am stuck in the coach and want to watch TV. Rain fade effects the dome frequently due to the reduced antenna size (lower initial signal strengths) and the additional moisture that can accumulate on the dome further reducing the signal strength.
Domes do have a convenience factor as you can't forget to stow the antenna so it's one last thing you have to think about before leaving, but otherwise it's much more limiting than the open face antennas.
The open face automatic antennas are the exact same antenna that DirecTV or Dish mount on a home so any and all hardware that you would or could use in your home can be used in your RV and work exactly as it would at home.