I've read through all the posts in this thread, and I'm much less confused now than I was earlier - so I want to thank everyone for their posts.
However my situation was somewhat different, because Norton Internet Security 2009 installed just fine with AntiBot already installed - no warnings or notices of any kind, however upon reboot NIS wouldn't run at all! 
So I tried to start NIS manually, by using the desktop icons and then trying the Start > Programs > Norton Internet Security > (and then selecting either Norton Internet Security, or Live Update) but neither of those would even start up - nothing. I looked at the Windows Task Manager to see if anything was happening in there, trying to figure out what was going on, but nothing.
It appeared that NIS was completely inoperative, dead in the water. I rebooted again, then I switched my user account back to having Administrator privileges again (after the first reboot, I'd changed my main user account back to its usual limited privileges), but that didn't make any difference either - the a/v and firewall part of the deal still would not even run, at all. However, AntiBot was humming along just fine. 
The only thing I could conclude was that AntiBot was preventing NIS from running, both at boot time and later when I was manually trying to open NIS. Temporarily disabling AntiBot made no difference, which had me worried.
So I uninstalled AntiBot, rebooted, and NIS seems to be running perfectly now. 
(Yes, I eventually changed back to a limited non-admin account - NIS is still fine - note that all installs were done using the same user account - all installs/uninstalls/everything were done under the same account, after changing that account's privileges with a separate Admin account - hard to explain without taking up too much space here. Anyway my earlier problem was not caused by something like installing under one account and running under another, if that would make any difference (some apps, it does) - nothing like that.)
Anyway, what prompted me to find this forum, and this thread, was to see if I even actually needed AntiBot or not, with NIS 2009, and it appears now that I probably don't need AntiBot.
Before reading this thread, I was worried that I would be exposing my computer to more risks, by uninstalling AntiBot, but it seems from what I've read here that NIS 2009 takes care of all that stuff already. 
Maybe I worry too much, but as fairly-new PC user (coming over from being a longtime Mac user and recent Linux user) I want all the security I can get on my PC, and I have confidence that Norton products can keep my PC safe. (No, I don't work for them; I just like their products, is all.)
Incidentally, aside from that little hiccup I mentioned above that prevented NIS 2009 from running, I've been very happy with Norton Internet Security 2009 so far (after a minor "Oh, no!" 
about the changes of the Log stuff, but I've adapted to that already) and NIS 2009 seems so much faster than what I'd been using before (I was previously using NIS 2007 plus AntiBot). I don't know if it was NIS 2007, or Antibot, or the two of them combined, that made everything less speedy, but I'm glad I moved up to the 2009 version of Norton Internet Security because it's much faster and much more user friendly IMO. (I like the new 2009 user-interface now that I've finally gotten used to it, although I didn't like it at first.)
- -
- On-again off-again Norton customer since 1990-something when they made their first Mac firewall.
The Norton-enabled PC is currently running:
Norton Internet Security 2009
Windows XP Pro SP2 (can't use SP3 'cause it consistently crashes this weird little Dell)
P4 Dell OptiPlex GX 260, 2.4 MHz
Edited, to add the following:
P.S. ... I did not use the Norton Removal tool, before installing NIS 2009 - perhaps I should have? Anyway, NIS 2009 is working fine now that I uninstalled AntiBot... I wonder if it would have made any difference if I'd used the removal tool... hmm.. well I guess I'll never know 'cause I'm not going to reinstall 2007, then 2009, just to find out. 