Limit:

Let S be a subset of a topological space X. A point x in X is a limit point of S if

every neighbourhood of x contains at least one point of S different from x itself.

Boundary:

Let S be a subset of a topological space X. The boundary of S is the set of points p of X

such that every neighbourhood of p contains at least one point of S and at least one point not of S.

So, deleted neighbourhoods of limit points must contain at least one point in S. But neighbourhoods

of boundary points must contain at least one point in S and one point not in S.

So, they are not the same.