When should I replace my companion?

Solution 1:

Both the Skyrim Wiki and UESP gives you a list of followers by the skills they tend to have.

Not all companions are the same. Some hate it when you break laws while others don't care if you go on a murder rampage right in front of them.

Every companion's stats is different since it depends on the level of the player when you meet them, but they all favor one of three types:

• Warriors like heavy armor, one/two handed, shield, etc
• Thieves favor one handed, archery, light armor, sneak, etc
• Mages favor spells over everything else

Here are some related questions for further reading:

• Who is the stealthiest follower?
• Do followers have skill levels and level similar to my characters?

Solution 2:

Is there really a reason to switch for a new companion?

Yes, companions do not advance. They are level-scaled to match your advancement when you enter their zone and then they gain nothing, ever. If you've stuck with the same companion for more than 15 levels, they're probably getting pretty weak.