The other part (which IMO is more consequential) is that once the LLM application quits or otherwise drops out of the alternate screen, that conversation is lost forever.

With the usual terminal mode, that history can outlive the Claude application, and considering many people keep their terminals running for days or sometimes even weeks at a time, that means having the convo in your scrollback buffer for a while.

>that conversation is lost forever.

You should be able to find it in ~/.claude

You can also ask Claude to search your history to answer questions about it.