What if any, is the difference between a band or a group?

Yeah, the two designations are probably more interchangeable than not. However (!), in spite of the use of the term “group” by even professional ensembles (hah!), I think that “band” tends to carry a little more weight as far as expectations of experience playing together and overall cohesion.  Number of members is generally irrelevant for designation as one or the other.

Then there is the term "boy bands". Were the Monkees an early arranged boy-band group?

Absolutely no distinction between a band and a group.  Although it's boy bands and girl groups, but that just the alliteration.  But then there's the Gang of 4 and the James Gang, both historical references.  But if you're looking for the ultimate expression of what it is, then you have to go with the Jimi Hendrix EXPERIENCE.

You can have a group of bands but you can't have a band of groups. Enjoy the music

Consider this: there are no bandies, but there are...