Why do I sometimes come across a label like v* instead of v in the literature? As I understood, it is the same thing.
They are not the same. 1. v* (with this label) corresponds to a verb projection with a full argument structure or what is called a Core Functional Category (CFC) with transitivity. This v*P is the small or light vP that can enter into a checking relation of Case/Agreement. Its head, along with C and T, constitutes the set of CFC. Also this head is always transitive in nature, it selects another V, it has phi-features for object agreement and case, and finally it has an optional EPP for object shift constructions. 2. The other v (the normal light or small v with no * label), however, is the one that Legate (2003) reacted to Chomsky (2000) for, saying that also passives, unaccusatives, TPs, other authors even suggested DPs, are phases on the basis of some phasehood tests she clearly shows in her article (based on wh-movement & Binding). To this point, Chomsky (2001) in another article has drawn a line between strong and weak phases, these weaks phases he considers passive/unaccusative vP, TPs, etc. They are not labelled v* because they don't have a fully argument structure. So indeed there's a difference. You could say that the light or small v (v*) is more functional in nature, and the normal light v is thematic or substantive in nature. This is briefly the difference between them as far as I know.