Which Omega theme should I use with Drupal 7? I need to use the Omega theme framework. I find there is two different versions: Omega 3 and Omega 4.

Which one should I use with Drupal 7?
There are Drupal 7 versions of both, so you can choose either. As of writing the 4.x branch is actively maintained, and 3.x not so much.
The 3.x branch will hold your hand a lot more, and make a lot of decisions about how you want to work. This may be or ideal, or may be frustrating. SASS support is clunky and a bit of a pain to work with, if I'm honest. Its main 'selling point' is that you can configure most of the options it supports through the UI.
The 4.x branch is much more of a developer-focused solution. Very little UI, but it incorporates a full front-end stack of tools for development/asset management. If you're a developer, this will likely be the one for you.
This is what the docs have to say:

Whether you should use Omega 4.x or Omega 3.x depends on several considerations:

Do you rely on the user interface for building layouts, or are you comfortable defining layouts in code in a tpl.php file? Omega 4.x does not include the elaborate layout definition UI that was a hallmark of Omega 3.x
Do you want to use a theme that is under active development, or are you comfortable with a theme that is minimally maintained? Omega 3.x is minimally maintained. In all likelihood, unless a new maintainer steps up to the task, Omega 3.x will not be upgraded to Drupal 8.
Are you comfortable with or willing to learn SASS, or do you want to stick with straight CSS? Omega 4.x does NOT require use of SASS but certainly is much easier to implement if you adopt SASS.


Beyond that, the only way you'll get a satisfactory answer is to install and try both.
Edit: to add a bit of balance to the above docs (thanks @FreeRadical for the info):
The 2nd point (claim?) is hotly disputed. It has been written by the Omega 4.x team who currently holds &quot;ownership&quot; over the project pages. This is a bitter dispute and there are negotiations currently taking place which probably will get the project page rewritten soon.
Sounds like fun :/