All these tools rely on Rhinocommon, my understanding is that
Weld Mesh uses
"Makes sure that faces sharing an edge and having a difference of normal greater than or equal to angleToleranceRadians share vertexes along that edge, vertex normals are averaged."
=> just a difference in rendering , suppress some creases
=> Could be useful if you want a cube look like a cube and not a sphere
Just a bit more powerful because it uses a tolerance.
=> there is a modification in Mesh topology, Very useful to seal the mesh, to lower the number of vertices, to connect faces, edges, vertices.
Join Meshes and Weld Edges
Join multiples separates meshes and may use
then weld all vertices near from each other.
For the normals you will also need Unify Normals but use it after the welding
If you go in C# it is important to follow these steps :
Optional : Unweld or weld depending if you want to see crease or not