So here's how I view it thus far:
NORMAL TAMES
+ Bonus levels with taming efficiency (which can be controlled, i.e. if you want a low level for breeding you can just punch them until 0%)
+ Can immediately saddle up, use, and/or breed
- Can be more difficult than just grabbing babies, especially for higher levels or tougher species
- If carrying with a flier, some can fight back so you'll need to watch your HP
BABY CLAIMS
+ Much easier to do, simply kill parents and claim baby
+ Baby dinos are cute
+ If you can carry them off with a flyer, they typically won't fight back
- Can't immediately ride/use/breed; baby needs to grow first
- Will need food on hand for baby so it doesn't die
- Not a guarantee to find one or even have one spawn at all
I could be missing a few points (and I'm still confused on the whole bonus level thing) but that should be the gist of it. Overall there are upsides and downsides to both routes. And, of course, taming the parents means you can claim the baby as well!
I adopted my first rex after seeing the parent killed by a few angry brontos while flying overhead, so that was definitely an easy powerful dino at the start. Meanwhile, I had to manually tame my first Theri because I've never seen one with a baby.