For what special cases does this vertex cover algorithm fail or work?
Trying to take "high priority" i.e. high degree vertices over their neighbors will not work in all cases. This algorithm chooses $n-1$ for $C_n$ (Cycle Graph on $n$ vertices). It should choose either $(n-1)/2$ or $n/2$ if its odd or even. It also fails for Perfect Graphs.

I'm pretty sure this gives a $O(\log n)$ approximation too. If you order vertices by degree and greedily select vertices until all edges are covered, it is basically the same greedy algorithm as $O(\log n)$ set cover.