Is a Band Without Its Original Members Still the Same Band?

That's a good example.
I think it's more about (& I've said this at HL too) a "signature". Something that identifies the band to its fanbase (& indeed anyone else who hears them).
I mean, The Doors really couldn't be The Doors without Morrison.
Just like The Sisters couldn't be The Sisters without Eldritch & The Doktor. Remove either & it just wouldn't work.

That's it. The "signature"(or "atmosphere") and the "face" (usually the frontman/singer) of a band have to remain intact for people (or fans) to still identify with said band. The bands I love I honestly couldn't imagine with a different singer, I certainly wouldn't even give them a chance but maybe that's just me. Lyrics are very important to me, also the vocals. It's not really about them being great singers in a technical sense but I have to like the voice of a singer.
Didn't AC/DC exchange their singer? I seem to remember something like that but am not sure, I don't know much about them.

To me it depends on a lot of factors and my comments and thoughts only apply to bands I care about, which I will discuss in depth here.. To me, it basically depends on what makes up the CORE of the band's sound.

In the case of The Cure, it's the vocals, the lyrics and the music that defines the core sound - not just the voice as "The Cure". I think that the musical direction has always been by RS. Sure the other members of the band have had an influence, and varying degrees of input in the song-writing, but really we don't know who has done what in which song unless we're told by the band in interviews and the like. Suffice to say though that without Lol and the others early on, The Cure we know today would not be here as it is now. In the case of The Cure, it seems that RS feeds off, or interacts with the other members to produce the music as we know it so as long as he is in the band, it's still The Cure. Look at The Top - from what we know that was basically a solo record (from the song writing perspective) but it is still "The Cure". Some might say that band members coming and going have a direct effect on the quality of the songs but I would disagree - to a point. In my opinion the self titled record and 4:13 Dream are the weakest albums the band has ever released and these records had two sets of different members. Wish and WMS had different personnel and sound vastly different but then you look at WMS and Bloodflowers both made with the same folks and the difference there is stunning... So IMHO, yes the band members do have an influence on the sound of The Cure, but ultimately it's RS who drives the end result. 

I have a different point of view regarding Jane's Addiction. Back when JA started the stories go that Perry (singer) would hum the basslines to the then bassist Eric Avery, and the two of them would create the songs from this. Later they found a guitarist and drummer, both of whom were replaced with Dave Navarro (guitar) and Stephen Perkins (drums) to make up Jane's Addiction (verison 1.0). JA's music whilst not bass-centric does rely heavily on basslines that are very much a defining part of the songs.... They put out a "live" album in 1987, and then two studio albums (1988, 1990) before breaking up in 1991. To me and other oldskool JA fans, these are the only songs that are any good. In the years between 1991 and 2002/3 Perry and the JA drummer started a new band - Porno For Pyros - which differed greatly from JA but a) used snippets that were played in JA songs that became PFP songs (Blood Rag, Orgasm) and b) they played some JA songs on each P4P tour throughout the years, even though Perry said he did not want to cash in on the legacy of JA :-) Perry's voice is unmistakable BUT the difference in the songs is easy to spot. JA had a metal/punk/jazz/slow insanity to it whereas PFP seemed a little less edgy and by their second album they had gone in a fully different direction to their first album. 

Years later (2002/3) JA reformed (again after a tour in 1997 with Flea on bass) but with the original P4P bassist as Eric turned down the offer to rejoin. The songs written for their new album "Strays" were just pure RAWK and did't have that arty vibe that the OG songs had - except for one song which was a rehash of an OG song that wasn't ever quite there and really didn't ever reach its potential... There was a reason it was dropped in the original days...

Fast forward to 2009 and JA is being recognised by NME for their God-Like whatever award.. Eric A is asked to join and he does, they play a few shows but the reason the band broke up is still there and even after a Trent Reznor led recording session AND an international tour with NIN and their own tour, Eric leaves the band again - basically because he can't stand what Perry has become. So then Duff McKagan (ex Guns n Roses) joins as bassist for a while, writes a couple of songs and then leaves the band... So the band meets with TV On The Radio's Dave Sitek comes into the studio with the remaining members of JA as their bassist for the songwriting ONLY. The result is The Great Escape Artist -  an album which is really terrible - from the perspective of what Jane's Addiction was... So after all that, what we have as JA today, in the eyes of a lot of OG fans is not the JA of old, and not because Perry's voice is shot, because of the man he is now and because the magical songwriting combo is no longer there.. So in this instance, it's not the same band because the music being made is not the same as it was originally. 

I could go on to discuss Faith No More (no Jim Martin & Chuck Mosely replaced with Mike Patton) and Tool (no Paul D'Amour) but I'm tired and I have typed out a tonne of words up there... I can go on and on about those bands if anyone wants, maybe tomorrow or another day...

thanks mate..
haha yeah Jane's have had a few bass players: Eric Avery (85-91) > Flea (97 Relapse tour) > Martyn LeNoble (2001) > Chris Chaney (2001-2009) > Eric Avery (2008-2010) > Duff McKagan (2010) > Dave Sitek (2011) > Chris Chaney (2012-present).

The Smashing Pumpkins is another band.. they've got their original drummer back now and Billy is playing a lot of old faves now..

Here's my like six years after it was written. Amazing post. Being an old school JA's fan, I feel exactly the same.

Thank you. Are you a member at aintnoroght.org?