The Undertones...

became That Petrol Emotion...

Personally I don't think either group is particularly great, but I seem to be in the minority, so...

became Doves without losing a single member, but definitely qualify because they changed identity after their studio burnt down, so were - *ahem* - literally 'from the ashes'.

then these things are so subjective...

Half of the original Human League became Heaven 17

Most of the original Adam & the Ants became Bow Wow Wow

We won't mention Sweet and the Wandering Crutchlees. Even I can't stretch the definition of 'great' that far.

became Amy Winehouse

Karl Wallinger left and became World Party, Anthony Thistlewaite stayed and produced a couple of thoroughly decent solo LPs (mainly about tractors)and the Waterboys kept on kept on going.
Fairport begat Steeleye Span and Richard and Linda Thompson and Matthews Southern Comfort and Fotheringay. Steeleye Span begat Gay and Terry Woods and the Etchingham Steam Band, the Albion Country Band, the Albion Dance Band and the Albion Band. (Am I stretching "great" too far here? I don't think so!)

Well I thought they were just as good.

Also At The Drive In became two bands that are almost, but not quite, equally as good, Sparta and The Mars Volta.

Also Mother Love Bone mostly became Pearl Jam and they were/are both great bands imo.

Finally of course The Small Faces became The Faces

OK, so while John Mayalls Bluesbreakers were still going,which wasn't the question, I know but, how about Cream, Fleetwood Mac, Colosseum for starters. The list goes way onward.
As for break ups, does Blind Faith into Delaney&Bonnie&friends, Traffic (take 2),Family and Airforce count?
10c.c. into, um, 10c.c. and Godley/Creme nearly made the grade until they realised neither were that much cop.
Beatles to Plastic Ono Band, Wings, Hare Krishna Temple and All Starr Band?

Spacemen 3 - Spiritualized - Spectrum

I also love the way that in my record collection they all sit next to each other alphabetically.

to Whitesnake, which I think had 3 members of the former band at one point?

Cream to Blind Faith.

Led Zeppelin to Page & Plant. Does that count?

The Jam into From the Jam so I'll nominate The Beat into Fine Young Cannibals

The Birthday Party morphing into Nick Cave & the Bad Seeds. Both Nick Cave and Mick Harvey made the transition

Split Enz -> Crowded House
The Move -> ELO

Family ... Streetwalkers, who were great.
Specials ... Funboy Three, who were not so great.
Sex Pistols ... The Professionals, who weren't at all great.

Became Bad Company.

For what its worth - Buffalo Springfield into CSN&Y (see what I did there!)

How about Linda Ronstadts backing band into The Eagles

Both seem to satisfy the "as good or better" test although I have doubts about the legitamacy of backing bands.

Byrds to Burritos?

Airplane to Starship? - probably fails the as good or better test though.

Great bands both.

Found this thread whilst googling McCarthy
The Yardbirds changed into a pretty good band...

becoming Soft Machine ( until Robert was gone ).

Became Heaven 17.

Whilst Human League still kept going.

And at the same time, Ultravox! Became Ultravox. Does losing the exclamation mark count as a name change?

3 ex members, Slash, Duff and Matt Sorum. Until their sad demise last year, they were a far better band than the ongoing Axl Rose fronted G'n'R project.

Into Arc
Into Bell + Arc
Into Blockheads (2 of them anyway)- probably with additions from Greatest Show On Earth.

Forevermore into Average White Band
via Glencoe etc.