Great question, the answer is much longer than what I can reasonably post in an SO answer.
If you are new to Android you should absolutely be using Android Studio. By using Eclipse you are going to be learning an outdated IDE (for Android specifically) which Google has strongly indicated they are not going to be supporting in the future. It is much better to learn the skill you are going to be using 2 years from now.
Also: Android Studio has come a long way and been updated multiple times. It was definitely more shaky in the past but I have now used it for two separate Android app projects without issue.
I have also written two comprehensive articles on this topic for anyone who wants the complete in-depth details. If you are still on the fence about which to use then you can read either:
If you just want a general overview of the differences:
http://www.airpair.com/android/android-studio-vs-eclipse
For Migrating from Eclipse:
http://rexstjohn.com/exporting-android-project-eclipse-android-studio-0-4-0/
