Technically, yes, they have overlapping purposes, and Mine Hunters are the past, while Countermeasure Ships are the future trend.
In practice, however, hunters tend to be more agile and maneuverable, made of non-magnetic steel, with tighter operational noise levels and special materials, designed to enter mined areas and keep them clean. While countermeasure vessels do the same but are made larger, less agile, classic materials and less complexity, more equipment and often keeps a distance to mined areas and use auxiliaries to reach.
