The two place the workpiece under different loads, and I would only use conventional when the workpiece, fixture and machine are not suited to the pulling forces generated in climb. A designer should keep these things in mind when designing fixtures, but of course, a lot of times you're making the best of what you have.
The only other place for conventional is when using negative ceramic inserts. Positive inserts should be used in climb.
For finishing, IMO, you should always climb. If the machine has no way of adequately resisting the pulling forces... then use conventional.
Conventional places a greater load against the cutter on exit when forming the chip, it creates more friction at the start of forming a chip and in some materials; high strength, low thermal transfer, cold work hardening, etc., conventional will impact cutter life significantly.
Whether to use carbide or HSS is a question of rigidity. If your machine and/or fixture isn't rigid and/or poorly damped, you may find yourself struggling with carbide edge retention. HSS is more forgiving but significantly less rigid.