Fine line describes technique
Fine line refers to the linework itself: thin, precise strokes. It is a technical description of how the tattoo is applied rather than a subject matter.
Minimalist describes composition
Minimalist describes how much is in the design. It is about restraint, negative space and reducing an idea to its essentials. A minimalist tattoo can use several line weights, though it usually does not.
They often overlap, but not always
Many minimalist tattoos are fine line, and many fine line tattoos are minimalist. But you can have a detailed fine line piece, and a bold minimalist piece. Being clear about which quality you care about helps the artist.
What ages well
Very fine detail can soften as skin ages. Minimalist designs with slightly heavier lines often hold their clarity longer. If longevity matters most, say so and we can adjust the drawing.