What is the difference between being "asked to resign" and fired?

They're probably doing you a favor by giving you the chance to resign rather than be fired. A resignation on your resume is much better than being fired.

With regard to unemployment, you can't collect unemployment if you quit (in most cases).

The probably wouldn't give you the chance to resign unless you had done something pretty bad. Under normal circumstances, if they lay off an employee, the employee can collect unemployment, and that will raise their unemployment comp expense. However, if you're fired "for cause," meaning stealing, harrassment, something like that, then they can contest the unemployment.

The probably wouldn't be offereing you the chance to resign unless they assumed you could be fired for cause (in other words they assume they'll win in denying unemployment comp).

They gave you a chance to resign,

they saved your job record from saying they fired you thank them

In reality, nothing. In pracitce, you get to say you resigned instead of having to tell future employers you were fired. Big advantage.

asked to resign: you voluntarily leave your job before being fired

Fired: boss fires you without you volunteering

being able to file unemployment insurance for one.