Graded papers have a set contrast level and are typically printed without the use of filters or gels. Grade 1 has less contrast than Grade 2, and so on and so forth. Grades 2 and 3 are the most common.Typically you would match your negative with a grade of your choosing for a particular effect. For example, if you had a somewhat flat negative, it might produce a correspondingly flat print at grade 2, so if you wanted your print to have more contrast, you would probably print Grade 3 or even 4 depending on how much you wanted.
Variable contrast papers have the ability to produce prints with varying levels of contrast, so instead of selecting a Grade 3 paper, you would use a variable contrast paper and print using a Grade 3 filter. You could use the same paper with a different grade filter to get a print that looked like a paper of that grade. The upside of VC papers is that you only "need" one paper since it's versatile, compared to needing multiple types of paper around to accommodate different kinds of negatives.
Some people simply prefer Graded papers, but they aren't necessarily inherently better.