Degree will open the door and maybe get the interview. I can't speak for everybody, rudee nailed that part, but my list when hiring is like this:
Education - The right degree means I look further down, but unless it's extremely specialized, it might not be enough to even get you an interview.
Experience - In your case, as a recent grad, I'd look for an indication that you've got some work ethic. Consulting on top of a full time job is good, having some solid summer work experience.
Above all else - give me a reason to want to meet you. There are lots of well educated people out there with some experience, so there needs to be something on the resume to pique my interest.
Software/IT isn't my area, but I imagine any technical position would be similar? Your resume needs to get you in the door, after that it's all you.
Hope it helps. Good luck.
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