Are there any problems in $APX - PTAS$ that are not $APX$-complete?
Yes, at least under some reasonable assumptions. Crescenzi, Kann, Silverstri &amp; Trevisan show that Minimum Bin Packing, Mininum Degree Spanning Tree and Minimum Edge Coloring are APX-intermediate unless the polynomial hierarchy collapses.

Considering that the paper is from 1996, I'm sure there's now a significantly larger number of known APX-intermediate problems.