Both Thresh and Morgana have their pros and cons.
Morgana Pros:
-Her bind has good damage + a long snare
-Shield is basically an anti CC against Leona + Thresh stuns (although I believe Thresh can still go in on you and flay)
-Ulti is mobile + can zhonays with it and still be active
-Taking the AP gold item (I forgot the name) is really effective with morg cause it's just easy af to proc it with her W.
-If you land a bind on an incoming jungler it makes the gank a lot harder for your opponents due to its long duration.
Cons:
-If you don't land your binds, you're pretty much useless except for your ult.
-Some champs have a pretty easy time getting out of her ult (Lee Sin, Ziggs, etc)
-If you time your shield incorrectly then it's also pretty much useless.
-As mentioned in the bullet above, you have to time well with Morg if you're against a good support who understands Morg's shield.
-Not really an "all in" type of champion who can just hard engage unless you flash ulti.
Thresh
Pros:
-Autos hurt with E passive
-YOU CAN GRAB HIS MOTHER FKING LANTERN
-His hook is also a slight stun
-If opponent is hooked you can time it appropriately to make the duration longer by pressing Q again and then flaying.
-Great disengage with E as well as ult
-Great engage with his Q-Q-R and then E the enemies into the walls.
Cons:
-Similar to Morg, if you miss your hooks it can be crucial due to the fact that your hooks are on a rather long CD.
-His lantern doesn't protect from CC but it compensates by allowing other champs to grab the lantern into safety.
-For some reason, people really don't like grabbing the lantern....
-Ulti isn't mobile
-It's very tempting to go all in as a Thresh but if you play it wrong, then it can be a crucial mistake for your team.
I think both supports are very strong in their own ways.
I personally prefer a Thresh support when I play adc because most people who suck at landing hooks are at least knowledgeable of this and therefore compensate for it by flaying then hooking.
However, if I play support I prefer Morgana because she can actually carry some games (essentially another AP Mid with the crazy damage output).
Playing Morg puts me in a better position to carry a game rather than Thresh who is imo better as a utility support champ.