I'd like to point you over to a great article over at Number One with a Bullard about what defines a band, and specifically whether a band with none of the original members is still the same band. This is a question I've concerned myself with many times over the years. Inspired by one of those conversations with Gabe, I literally asked the same question about The Smashing Pumpkins and Zwan when I saw the act billed as the former in 2012. It came up again when I saw Tangerine Dream a few years ago, and in that case it did not at all negatively affect my enjoyment of the performance.
Anyway, Gabe's been writing on all sorts of topics in media and culture relevant to my interests, asking and answering plenty of questions that have rolled around in my head as well. Go check it out. There's a written version and a podcast version (at least for the newer posts), and they're usually very similar but sometimes have slight differences. It's all worth it.