Hello!
The meanings of these two words are very similar:
A boundary is a line that marks the limits of an area; a dividing line - this can be a wall, a river, or a line.
A border is a line separating two countries, administrative divisions.
But
I would say that "boundary" could be used to refer to the line around (or between) anything like an office, a piece of land, or a garden.
and
I would say that "border" is used more to describe the line dividing two countries, or legislative areas (like the states or provinces within a country).
I hope that helps!