Which one of these programs would be considered more robust? In addition, which one has the steeper learning curb?

Not sure what you mean by "robust", but each has some advantages. If you are getting just Premiere and not the production Suite, Vegas will have a great many advantages. If you are comparing Vegas to the Adobe Production Suite, then it's no contest.

In my experience, if all you're doing is editing and you're new to it, then Sony Vegas is probably the better option. It's slightly easier to learn. Please note that complexity in software has its own reasons.

However, if you need a range of services, like compositing, image processing, motion graphics and authoring, then nothing beats the Adobe Suite for value and quality.

Both are robust enough for professional work.

I think the two replies so far reflect a good summation of previous similar posts so let's take a pause...

Variations of these very subjective and general questions get posted about every month or so and the threads usually become rancorous. Before that happens, I'd suggest to read through the two respective forums, download the free trials for a test drive, and then post any more specific questions that haven't been already addressed.