A lot of it is down to personal preference! If there is a NINO group having meetings near you (nineinnineout.org), you can try on different carriers there and decide for yourself. I didn't try a mei tai or wrap until my daughter was three months or so, so you can figure she had some head control not like a newborn, and then I did only front carries for a while. I started back carries when she was sitting well on her own (you can start earlier of course.)
So, with that in mind, I preferred a narrow or small-bodied mei tai rather than a large-bodied one when she was little because she did not like to have her legs "in". I tried a Freehand (small) and later a Kozy (large)... I still have a Freehand but don't have a Kozy (lots of people do like them though). I've heard of people using either with newborns - since the baby is smaller to start with, the height of the MT does not matter so much to ensure support at that age (it's more of a thing later when they are taller and napping in it.)
I also liked using a wrap in "front cross carry", starting in the autumn - it seemed like too hot a thing to try in the summer. I used an Ellaroo and a Gypsy Mama BBB (I still like these brands if it is hot out, they're thinner). For head support I could just tuck her head in on one side of the cross. When I was ready to try back carries in a wrap, I got an Inka Storch because I had heard that a Storch was great for rucksacks. I did not really like the Inka though (personal preference here, a lot of people like Inka but I didn't really like the texture or color). Towards winter, I got a Didymos - I like those 
and I also like Easycare and Leo Storch - for a heavy child (18 months and counting) I find any of those to be fine. They're all thicker and heavier-feeling than an Ellaroo or Gypsy Mama would be.
A mei tai is easier to learn to use and cooler in summer and maybe more fashionable looking (or maybe I'm just deluding myself). A wrap feels more comfortable and supportive to me now.