Pretty much the title. I keep reading the definitions but both concepts seem to be the same thing. At minimum in the very basic examples I have currently worked with (just starting to learn about the topic).
Definitions:
The boundary is the closure of the set of all simplices σ that are proper faces of exactly one simplex of K′. This definition naturally captures what you might think of as the “boundary” of a set.
The link $Lk(S)$ is equal to $Cl(St(S)) - St(Cl(S))$
I assume K' is an abstract simplicial complex.
Boundaries are only well defined for pure simplicial complexes. Importantly, the boundary of a simplicial complex is a subcomplex (it's contained within the set you are considering).
The link gets the boundary of the neighboring simplices. It's not contained within the set you are considering at all.
For examples, consider a pure simplicial 2-complex that is a manifold. The link of a vertex will be topologically equivalent to a 1-sphere. The boundary of a vertex is the null set.