I started to get into the NOS tube-swapping thing a while back, but I quickly got tired of it...
In my experience, YES- different tubes make a difference in tone, but not enough to justify some of the big bucks people spend on NOS tubes. I do think that some of those old tubes tend to last longer though. Keep in mind that different tubes sound "different", not necessarily "better".
The effect on tone depends on the position of the tube. In a Twin, I think you would expect to see the biggest difference in tone from the V1, V2 & V3 preamp tubes. If you use alot of reverb, whichever tube is the reverb driver will have a big effect on the sound of the reverb... The power tubes can also effect overall tone and can give you more or less clean headroom...
If you pull two output tubes then you have to be sure to cut the impedance to 1/2. Check the manual on that Twin reissue, it may tell you exactly how to do it. I've been running an Evil Twin on two output tubes for quite awhile...
As for tubes, I like JJs. They're kinda dark or "warm" sounding to me. Whomever you buy them from, just make sure they are tested. KCA Tubes is very reputable, I would recommend them. He specializes in NOS stuff, but he has the better new production stuff too...