What is the difference between Transitive Closure and Join?
No, it is not. Transitive closure is the closure of composition on binary relations; composition can be expressed as a rename (to make join operate on the right attributes), followed by a join, followed by a project to remove the common attributes.

So composition can be expressed in terms of join, but (as Erwin Smout says) its transitive closure can not, and join (or composition) cannot be expressed in terms of transitive closure, either.