What are the advantages and disadvantages of terrain and mesh in ORD?

mesh is general 3D "fabric".  Terrain is a specialization of a mesh, and it has special powers, like built-in contour display.  Certain tools work with terrains, but not meshes (especially drainage tools).  Some tools build meshes automatically, but not terrains.

A terrain is a 3D planar element, but it does not have a depth/thickness. Think of it like a piece of paper that has points jutting up and down on it. A mesh, on the other hand, is a 3D volumetric element that does have a depth/thickness. Think of common 3D shapes, such as bricks/blocks. 

You can calculate the volume of a mesh, but you cannot calculate the volume of a terrain by itself. This is why there are terrain to terrain volume tools. In this case, it is essentially making an (invisible) mesh and then measuring the volume of that. 

A terrain cannot have vertical surfaces either, but a mesh can. Because of the way a terrain triangulates, there needs to be some horizontal difference between points, even if very small (for example 0.01 ft). 

I just wanted to add a few thoughts and clarifications here.

I like to think of a Terrain Model (TM) as a specialty mesh element. A TM has a couple of specific requirements and behaviors as Eddie was alluding to, no vertical surface or overhangs for example. A TM is going to be a triangulated mesh (this is where we get the language of a TIN, a Triangulated Irregular Network), that uses a specific algorithm to define the triangles. Most civil design applications us a Delaunay triangulation to define this and will often allow the user to modify triangles to meet specific needs. As Jeff mentioned, ORD has specific tools and settings that are intended for use with TMs that relate to our workflows as civil designers, contour generation, Active TM, specific sampling tools, etc.

Meshes are networks of faces, usually in 3D that are defined by polygons (usually 3 or 4 sides). As three-dimensional objects these can have vertical faces, "overhangs" and be either open or closed. An open mesh is the "paper" concept articulated earlier while a closed mesh would enclose a volume (it is this that can define a volume. 

The nature of the elements points to their specific advantages/disadvantages as others have indicated. As a general rule, I interact with TM in my efforts with the exception of some specific use cases (think cut and fill volumes which are generated as closed meshes). You would have to expand on the workflow you are referencing for me to be able to comment on the need to convert TMs to meshes.

Steve