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

My personal feeling is a band consists of 4 or more artists. A group, 3 or less.

Any thoughts if this is a distinction that has any merit?
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Seems there’s a mixing of terms such that the definitions are blurred. We could start with: Everyone in a band plays instruments or is a musician. Not everyone in a group does or is.
A group, 3 or less.
I would call The Mamas & the Papas a group (4 members)
A band holds things together. A group is more than one. Got it?
The Mama’s and the Papas weren’t the norm though, were they?

"Got it?" = another know-it-all answer from our resident genius? No wonder first time posters are more than put off.
Spencer Davis Group - 4 musicians.
Just vocalists...it's a group.  If there are musical instruments involved, it's either a band or a group. 
The Jeff Beck Group......isn't The Jeff Beck Band.
The Band....isn't The Group.
Hendrix' Band of Gypsies......isn't Group of Gypsies.
The Allman Brothers Band....isn't The Allman Brothers Group.
Looks to me like names are chosen by how they sound when spoken rather than on other criteria. Personally, I think of a "band" as a "group" of musicians who gather and play on special occasions or in parades; and I think of a "group" as a small collection of performers who perform for entertainment......and money. But I could be wrong.
No absolute difference, depends how music industry would call them.
     A group of pandas is called an embarrassment.  I can vouch for the appropriateness of this name after taking 4 of them to Chucky Cheeses for Elmo's birthday. 
      Three or more geese flocking on the ground is a gaggle; three or more geese flocking in the air are a skein. Here are a few more off the top of my head:
  • Cattle: mob or herd
  • Cheetas: coalition
  • Colts: rag or rake
  • Deer: herd or parcel
  • Dogs: litter (if they're puppies), pack (if they're wild), or cowardice (if they're curs)
  • Dolphins: pod
  • Donkeys: pace
  • Elephants: herd, parade, or memory
  • Elk: gang
  • Ferrets: business, hob (male), jill (female), kit (babies)
  • Foxes: leash, skulk, or earth (this is the oddest of all group names in my opinion)
  • Giraffes: tower
  • Gnus: implausibility
  • Goats: trip, drove, herd, flock, or tribe
  • Gorillas: troop or band
  • Hedgehogs: array
  • Hippopotamuses: thunder or bloat
  • Hyenas: clan or cackle
     Oh, did you mean musical groups and bands?
How about this?
A band strongly implies that the members play instruments, aside from just singing and dancing. The Beatles and U2 are examples of bands. A group can be any band. The Beatles and U2 are also groups. But a group can also refer to performers who don’t play instruments onstage (or in a music video). These performers primarily sing and/or dance.
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Would groupies be considered 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...
yogi boy -- I think you got it spot on.
With all due respect...nothing. It's just interchangeable terminology.
Someone tell me what a combo is.
More than one topping.
Completely interchangeable. Neither has any relevance other than preference of word. I’ve been in bands, and groups and I’ve been in bands that we sometimes referred to as a group or vice versa. Not complicated.

Having said that, a “band” is loosely referred to when the members play instruments. Rarely is a vocals only “group” referred to as a “band”. That would maybe be the only real distinction. But plenty of “bands” are groups...
A combo is a small jazz group.
Nobody ever will get it I guess. A group is a bunch is more than one. A band is whatever holds them together. Band of brothers. Not group. Band. What holds them together? A bond. What bond? Brotherhood. Band and bond are similar. Band and group not similar, but related. Words you see actually do have meaning. Even though lost on many who use them.
Big Band Width, my waistline.
Super Group, the crowd waiting to eat at Yez Kassem's.

I think yogiboy had it the closest. It's a set theory thing. All bands are groups but not all groups are bands. All music groups make music but to be a band, you have to play instruments. If they don't play instruments, they are a vocal group. Boy bands is a misnomer and was created because it's catchy. 

When The British Invasion swept across the U.S., The Beatles, Stones, Dave Clark Five, Animals, Manfred Mann, Gerry & The Pacemakers, Spencer Davis (yup) Group, Yardbirds, Who, Small Faces, etc. etc., were called Groups. The UK press called them Beat Groups. As already mentioned above, Jeff Beck's first solo album was by The Jeff Beck Group.

I have long made the distinction thusly: If a musical ensemble is self-contained (writing the songs, playing the instruments, and doing all the singing), I think of them as a Group. I reserve the nomenclature Band for a musical ensemble which provides musical accompaniment for a singer or group of singers. For example, Frank Sinatra with The Count Basie Band (also sometimes named Orchestra). Frank was also accompanied by The Nelson Riddle Orchestra. In middle and high school, we had both a Band and an Orchestra. The Orchestra included strings and woodwinds, the Band was primarily brass and percussion. 

But by the late-60's, the term Band started coming into use in Rock 'n' Roll. While providing accompaniment for Bob Dylan on his 1965-6 world tour, The Hawks were definitely serving as his Band. When their first album was released they became known as The Band (not by their own choosing: contemplating a new name to replace The Hawks, they considered The Crackers or The Honkies. Capitol Records actually chose the name The Band on their own!). 

When Rock 'n' Roll criticism became "adult" (In Crawdaddy, Rolling Stone, regional alternative papers), the kids in the suburbs who formed Groups in the wake of The British Invasion---and produced regional and even some national hit singles---became known as Garage Bands. The Chocolate Watchband, The Syndicate Of Sound, People, Stained Glass, The Count Five were all San Jose Garage Bands I saw live in '65 through '68. I even saw Stevie Nicks and Lindsey Buckingham in their San Jose Garage band Fritz (shortened from The Fritz Reiner Memorial Band after receiving a threat from the great conductors estate ;-) .

But as the old expression goes, what's in a name?