Mesh is not a perfect world. Some of the problems relate to the fact that it is so new and its capacities will grow with time. For instance, at the moment mesh avatar creators are confined to the armature (skeleton) underlying the basic SL avatar. This skeleton is manipulable to a degree, but if you want to be a mesh spider you may have to wait a while.
1. Adding a script to a mesh object will make the prim count explode. This is based on the fact that a script places a heavy burden on SL's servers. This is true of any scripted object. The controversy that has arisen is that LL chooses not to represent this burden when the scripted object is made of regular prims and/or sculpties, but have chosen to burden mesh. Weighting mesh in this way will likely kill the possibility of certain mesh vehicles and many, many interesting mesh builds and devices.
2. The larger the mesh, the higher the prim count. This is another rather dreadful choice from LL, and is a terrific disappointment to building and tree creators. We finally have a prim size increase to 64m and gorgeous big builds are going to be punished. There may be a balance to be had, cutting larger mesh objects into smaller pieces and finding a prim count "sweet spot," but the purpose of having larger prims doesn't apply to mesh so much as creators may have hoped.*
3. The great ability to make a mesh have physical properties, like being able to walk through doorways and walk on floors? Right, that increases the prim count. Want to keep the prim count down? Don't define a physics shape for your mesh, set your mesh to phantom, and define the physics shape with regular prims (get a good 100% transparent alpha texture— you'll need it).
4. Mesh is one-sided. Imagine looking inside a shape that has a regular texture on the outside and a 100% alpha on the inside. This is due to something called "normals" that tell the system which way light is reflecting off an object and yielding an image. At present they can only go one way. If you want two sides, you have to make two sides. This, of course, increases the prim count.
5. Linking increases prim count. This one is rather infuriating, because mesh is set up to practically require linking to regular prims (see #3). But link a mesh into your regular prim build and watch the prim count of the whole thing go up. Building creators will likely be stuck having to use packaging systems, just as they do when using phantom prims in a build. However, see #1. Rezzing a compound object with the rezzing script in the mesh for final positioning may cause a temporary prim count so high that your customer can't rez the build.
5. Mesh cannot be flexible.
6. Mesh avatar faces do not animate.
7. Creating mesh requires use of an outside 3D modeling program. I use Blender. Learning Blender is *#&$@&$*@#.
8. Turning to the very dark side, the potential for vast quantities of mesh to be uploaded into SL from outside libraries is huge. There might be massive intellectual property law violation ahead. Yes, you have to have payment information on file with SL before you upload mesh (so they know who you are), and you have to pass a dorky little quiz about IP issues written by LL's lawyers to protect them from exposure before you can upload mesh (hint— you can't fail the quiz). But there are an astonishing number of ways around these flimsy barriers. And that's not to mention the copybot exploits within SL itself, which are an unaddressed plague.
*Don't bother trying to upload a mesh at a tiny scale and then scaling up. Doesn't work. But do be ready to see the prim count of your mesh increase as it grows larger.