They've each got their plusses and minuses.
Lockon/Chain on:
Pros:
Can be used in trees that aren't totally straight and limbfree
Can be locked in place and left behind without too much worry of theft
Cons:
Harder to set up initially
Comfort is usually less than a climber
Usually no rail around it.
A bit harder to use the lineman's style harness all the way up. At some point, when transitioning into the stand from the steps you'll be off harness for a minute, unless someone else has figured this out better than me, yet, and if so please elaborate.
Climber:
Pros:
Easy to set up and get up the tree
Comfortable for all day hunting (I've sat in one for 8 plus hours before)
Usually has a rail around it for improvised rifle rest, security/safety, etc.
Easier to use a lineman's style harness from the time your feet leave the ground till the time you return.
Cons:
Easier to steal if you decide to cable it to the tree and leave it behind (I usually do this though).
Usually heavier, though there are some nice lightweight models out there.
I have several of each and use them in different areas and situations. Some locations require one or the other, depending on circumstances.