Just to start... I'm not a fan of the elliptical. So, my opinion may be biased - but that is what an opinion is, right?
Anyway, I prefer the exercise bike because it is a non-weightbearing, no impact exercise. This is great especially for people with knee or leg problems. You get great development of all your leg muscles (including your gleuts!), which in turn gives you better strength in your joints. I also find that I naturally tighten my abs as I increase the intensity, and hit my abs just a tad as well.
I don't care for the elliptical, because I find it took longer to see/feel results, I don't feel the burn as well, and it's easy to cheat.
Whatever you choose, be aware that the ABSOLUTE BEST time to do your cardio is in the morning before breakfast. This will jump start your metabolism for the rest of the day.
Good luck on your fitness goals!