Like with living primates, we are able to definitively distinguish catarrhines from platyrrhines in the fossil record by what feature

The structure of their noses.

Explanation:

Catarrhines are distinguished from platyrrhines by several traits but the main distinctive feature between them, is the structure of their noses. This forms the basis of their names.

Platyrrhines are characterized by laterally-placed, rounded nostrils or flat-nosed while catarrhines have narrow, downward-facing nostrils or hooked-nosed.

Another distinguishing feature include their dental formulation which is 2.1.2.3 in catarrhines and 2.1.3.3 in platyrrhines.

The skull structure of catarrhines have frontal bone which make contact with the sphenoid bone unlike the platyrrhines.

Generally, catarrhines are much bigger in size than platyrrhines.