I haven't rolled tubes in pedals, but I have rolled quite a few in preamps.
The tube interacts with the circuit it is in. Some circuits make it fairly easy to hear differences between different models of tubes. Other circuits pretty much make all tubes sound the same; so at most you may hear a subtle difference in gain.
Differences you may observe:
1. Variance in the tonal contour.
2. Variance in gain.
3. Variance in how the tubes transition from clean to compressed to overdriven.
4. Variance in the tone or quality of the OD
5. Variance in distortion or harmonic content when the tube is run clean.
@BassikBrad has done some response plots of various tubes he has rolled. The surprising thing is the tone contour can be almost identical between two tubes that sound like they have a noticeably different tonal balance. My guess is part of the difference may be in how the tubes respond dynamically to different frequency ranges.
It's true that 12AX7, 12AT7, and 12AU7 all fit in the same socket. It's also true that they have different voltage gain factors. From a technical standpoint the circuit should be optimized to run a specific tube type. Tube are usually biased to the center of their operation range. 12AX7 and 12AU7 in particular have significantly different bias points. So 12AU7 may not work particularly well in a particular circuit set up for 12AX7. In most instances I believe there is little risk of damage though.
Also there is some degree or matching the tube type to the circuit. In particular, 12AU7 is probably a better choice for cathode followers because it has a much higher current capacity than 12AX7. Keep in mind the gain factor of a cathode follower is <1. Also this is not a recommendation to put 12AU7 in cathode follower circuits that are setup to run other tube types.
People often swap a lower gain tube into a circuit with the idea that they get more clean headroom. In most circumstance this not what happens. Instead you may be able to turn the gain up higher before distortion sets in, but the circuit will actually have less clean headroom. Unfortunately a lot of people interpret what there eyes are telling them to mean the amp has more clean headroom.
Swapping tube types may still be useful if your actual goal is to allow the gain control to operate in a higher range where it works better. For example, 12AT7 is the original tube used as a Reverb Driver in many vintage Fender amps. Unfortunately the Reverb control can be very touchy when the amp has a 12AT7 reverb driver. It's fairly common to throw in 12AU7, which allows you to run the control higher in a higher range. In my experience the reverb is changed subtly as well, which some like and others do not.
I have rolled a majority of the current production 12AX7 types. I do not have much experience subbing in different tube types. My observation is the 12AX7 seems to have more tonal variation than the other tube types. For bass amps my favorite combination is a current production Mullard ECC83 in V1 and JJ ECC803S pretty much everywhere else. But this combination is not always what an amp needs to give me the sound I want. Also you may hate this combination, so roll our own.
JJ ECC83S tends to sound way different than the JJ ECC803S. My perception is the ECC83S is relatively low gain and focuses on the mids. It does not seem to have much extension in the lows or highs, so to me it seems like it has a narrower bandwidth than other tubes. It also seems to have a bit of grainy texture in the high mids, like it is always on the verge of breaking up. In every instance where I rolled tubes in a bass amp, the JJ ECC83Ss came out. I have recycled some of them in my a few of my guitar amps though. Also some people love the JJ ECC83S in bass amps.
To my ears the JJ ECC803S has a wider bandwidth. The lows are somewhat lean, but focused, muscular, and clear. Rather than having a texture, this tube seems to add what I call a harmonic shimmer in the upper mids. To me I perceive the harmonic shimmer as color rather than a texture. I does not add gank in the high mids.
If you like gank in the high mids, try a current production Tung Sol 12AX7. This is probably the highest rated 12AX7 on the market, and I can understand why people love it. But it tends to be very punchy and hard sounding to my ears, especially in the high mids where I like a bit of a dip and maybe some compression.
I could potentially see a use for a Tung Sol 12AX7 in V1 and and JJ ECC83S in V2. The Tung Sol is clean and seems to be above average gain; so it could be useful if you want to push V2 into distortion.
In some circuits the tonal difference between the tubes I have mentioned can be very noticeable. In other circuits they all sound pretty much the same.
So in my experience tubes do seem to have sort of a personality, but the degree in which this personality emerges varies by the circuit the tube is run in. Also, the personality may vary somewhat across circuits.
The changes a device needs to produce the sound you want can vary from one device to the next. So a Tung Sol may be the perfect tube for one device and awful in the next.
The ideal tube can also depend on how you intend to use the circuit. For example I rolled tubes in one of my amp's OD circuits and the standard Gold Lion ECC83 (Chinese) was my favorite. However, so far I have found that I prefer the Russian ECC83/B759 in circuits that are run clean.
The tube you ultimately choose may require you to decide which quality you find more important. The first time I rolled tubes I found it really hard to choose between the current product Mullard ECC83 and the Gold Lion ECC83/B759 for V1. Both sounded fabulous but for different reasons.
The Gold Lion had awesome mids with sort of a vocal quality, but when I turned up the lows they became sort of loose and woolly. The Mullard produced heavier lows that were tight and focused. I choose the Mullard. I felt that I could crank up the mids on the Mullard and get 95% to the Gold Lions magic. But if I cranked up the lows on the Gold Lion, the sound was more like a Sovtek 12AX7LPS.
I have always been hyper focused on tone. I would say in most instances the change you will produce by swapping tubes tends to be rather subtle. Some will not be able to discern the difference, and some will not hear a difference that is significant enough to warrant the cost of the tube(s). Conversely you may find that a subtle change is all you need to go from hating a device to loving it.
IMHO, if you are interested and can afford it, roll some tubes and make your own decisions. You may find that it can be very worthwhile, or that it is a complete waste of time and money.
I suggest that you roll the various tubes you have through each circuit. In other words find the specific tube you like in each tube position, rather than taking a shotgun approach where you replace all tubes at once.
Also, consider going through a second and possibly third round of tube rolling. The idea is you may find a certain gestalt from a specific combination of tubes. You may not find this combination if you accept the choices you made in the first round of tube rolling.
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