Here's a rare category I just thought of. Two different bands who cut professional releases and who happened to share the exact same moniker. I can currently offer these four examples:

KALIDESCOPE:
Both were bands of the late sixties who played physchedelic-style rock. One was more ecclectically eastern-tinged in the George Harrison style, while the other embraced Strawberry Alarm Clock-style groovy pop.

KING HARVEST:
Two groups in the seventies shared this name. One based in New York was famous for a hit single called "Dancing in the Moonlight". Another, based in Melbourne, Australia, was known for a hauntingly beautiful cover of Jimmy Webb's "Wichita Lineman" which also incorporated "By The Time I Get To Phoenix".

NIRVANA:
The original Nirvana was another late sixties physch-pop band who jumped on the rock opera bandwagon with a sci-fi themed album called "The Story of Simon Simpoath". Oh yeah, I think there was also another band called Nirvana in the nineties . . .

SURVIVOR:
The first Survivor was a Louisiana based hard-rocking outfit with just one album release in their brief career. After their demise, another band rose from the ashes to carry the Survivor mantle . . . becoming an overnight success story with their song "Eye of the Tiger" becoming the theme for the motion picture Rocky III, they went on to be one of the top FM-oriented bands of the eighties.

Space:
French instrumental group of the late 1970s who are most remembered for Magic Fly, a UK Number 2 in 1977. Also a late 1990s group from Liverpool best remembered (by me) for the song 'Female Of The Species' & album Tim Planet.

The Eagles:
In addition to the Eagles that we're all familiar with, there was a short-lived British band by the same name in the early 60s. I think they did mostly instrumental rock without much vocals.

Slaughter:
Slaughter sounds like it should be the name of a menacing heavy metal band, and it was indeed the name of an obscure Canadian thrash band. Most people, however associate the name with a pop-metal hair band of the early 90s ("Up all night, sleep all day")

THE NAZZ:

The original version of the Alice Cooper band went by this name, and cut one single under it. However, when Todd Rundgren's Nazz emerged and became more successful, they changed their name.

ANDROMEDA:

Andromeda is currently a Swedish progressive metal band. Thirty years before this band was founded, another hard-rocking prog outfit carried the name, led by future Atomic Rooster star John Du Cann. This band cut one memorable album before disbanding.

Okay, I found out today after browsing a book on the physchedelic (a word I'll someday learn to spell correctly) era that there were not two, but THREE bands of that late sixties era to carry the moniker Kaledioscope (another word I need to learn how to spell - just call me the Stephen Ratliff of music writing and remember that all these errors are of course to be "ingored").
The third band, in addition to the two previously mentioned, came out of Mexico and released one LP in '69.

"psychedelic".

THOR'S HAMMER:

There was a Seattle based band who recorded a "doom metal" (whatever that is) album in the mid-nineties under this name, which is considered a classic of the genre and has recently been re-released. There was also a band called Thor's Hammer from Iceland in the late sixties who had their smash hard-rocking single "My Life" immortalized by inclusion on the Nuggets II box set.

GYPSY:

Two different "Gypsies" from the same era, playing the same style of rock music. One from the US (Southern California) and the other from the UK (Leicester). Both were heavily influenced by the "San Francisco sound" of simple folk-blues-rock incorporating improvisational free-form long jamming such as that pioneered by the Grateful Dead, Jefferson Airplane, Santana, et. al. Both were total commercial unkowns and failures in their time whose albums have since achieved the status among record geeks as prime collectibles.

Believe it or not, there were actually two bands called The Velvet Underground! Aside from the one we all know, the All Music Guide to Rock mentions an Australian band by that name appearing on a compilation of Love covers.

Also, speaking of early Alice Cooper/Nazz, they started out as The Spiders (releasing the very cool "Don't Blow Your Mind") "The Spiders" was also the name of one of Japan's biggest indigenous rock bands in the late '60s.

Here's a couple:

The Firm - One was the "Radioactive" one featuring Paul Rodgers and Jimmy Page. The other went "Star Trekkin'" ("Across the universe/On the starship Enterprise/Under Captain Kirk).

(The) Frantics - One was a '60s-'70s garage rock band. I've heard one song by them, "Werewolf", an interesting instrumental. Then in the Nineties, there was a Canadian comedy troop by that name. Their most famous bit was "Tae Kwon Leap - Boot to the Head". (They've since reformed with some hilarious new material. "Photo of Mary" and "American Jesus".)

"Music is a world within itself and a language we all understand."

Not to nitpick, but there were actually TWO bands going by "The Frantics" in the '60, who actually seem to get confused with each other quite a bit (even AMG blurs the two bands into one entry). The guys who did "Werewolf" were from Seatle and were active from the late '50s to the early '60s. On the other hand, there was a late '60s, Psychedelic "Frantics" who started in Montana and roamed all over the palce before breaking in in '71 (their album Relax Your Mind is very cool). As I mentioned before, AMG seems to have unknowingly included both bands' recordings under a single shared name.

AMG also does the same thing with the two "King Harvest"s I mentioned in an above post. It seems to think the band that did "Dancing In The Moonlight" is the same band that covered the Jimmy Webb tune.