Which software tool is widely preferred around the world for finite element analysis, "Nastran & Patran" or "Ansys"?

Patran (Pre & Post Processor) + Nastran (solver) is used in Aviation Industry. It works best with thinner surfaces. Still valid enough for all the general application.

Ansys is used every where esp in Machine tool, Automobile etc. Ansys acquired many companys and combined they give solutions to a wide range of applications including flow analysis, hydrodynamics, etc. They are got better GUI compared to Nastran.

Mastering any of this tool is good. Both are equally good. Most important is improving our knowledge esp on multiple features in it.

for aerospace we should choose patran and nastran

for automobile we can choose ansys

but out of these two,  nastran&patran is best

Depends on which industry you are looking at & the type of analysis you intend to do.

Generally in Automotive Industry Abaqus (Good for Non Linear problems involving contacts), Nastran, LsDyna (Explicit solver for Safety), Radioss, Madymo are the popular ones. Aerospace industry also Uses Abaqus & Nastran widely. I have seen Companies involved in heavy engineering & Part suppliers using more of Ansys & Nastran.

Additionally; for CFD there are another set of software's like FLUENT, StarCD, AVL Fire, Excite etc.