This is not a ubb issue but a Apache issue.
Never change the case on a ubb script file because Apache will not find it.
Assuming you have the default ubbt prefix in your MYSQL which it looks like you do.
Looking at the first table it should look like:
ubbt_ADDRESS_BOOK
then the next should be
ubbt_ADMIN_LOG
And so on.
I don't even want to get into asp pages. I used them, two years later I am still going through hoops to fix redirects.

As far as external files the habit to use lower case is not the issue.
say you made a home page index.html that is fine.
But if you add a reference to that page somewhere else say in a island and you typed Index.html. Windows will work fine.
But Apache will bark back and say the file does not exist. Because the first letter is upper case.
Also If you have a home page and your url goes to say Ubbthreads.php instead of ubbthreads.php it will not work because of Apache thinks it is a different file.
The funny part in my previous example about the three index.html files. You can have all three in the same folder but when you download a backup it will just overwrite each one.
Just a FYI

I think that all of our links are lowercase.

As for the databases, the names should be uppercase (where your table prefix would be whatever you have set in your config file)

Sorry, guys. I'm getting killed here... been trying for over a day to get the links fixed. This is amazing since I've run a UBB site for over 8 years!!! Ouch.

Here's what I transferred from the old site. (name changed to olddomain.)

'FULL_PATH' => '\\Inetpub\\vhosts\\olddomain.org\\httpdocs\\forum',

AND THE ERROR THAT I GET
Failed opening required '/Inetpub/vhosts/newdomain.org/httpdocs/forum/libs/functions_forums.inc.php' (include_path='.:')

I've changed every "\\" to a "/" and it still doesn't work. Also changed just one at a time and still no workee.

It seems to me that two "\" equals one. Right?

Any ideas what I can try next?

Oh, what the heck... here's the current link:
http://vcca.org/forum/

I doubt that path is correct; I highly doubt your new host has inetpub since it's a linux host... you'll want to get the new full path from the host.

They'll likely give you the full path to the directory under your web root (where you end up when you login via a FTP client) from here you'd have to direct it to your ubb install.

Example, if they tell you:
/home/yoursite.tld/

And when you login to your FTP you see "public_html" and you have a "forum" directory under that, then your UBB directory would be:
/home/yoursite.tld/public_html/forum

Yep.. That was it... dang... so obvious... Linux (NOT Windows)
For the record the path was:

/var/www/vhosts/domain.com/httpdocs