Great question. I hope someone knows the answer!
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Based on a review I've read, one difference is the amount of gain each of these driver tubes provides. The Ray Samuels Emmeline II Stealth preamp will accept any of these tubes without any other adjustment, thereby altering the amount of gain produced. The review is at http://www.6moons.com/audioreviews/emmeline2/stealth.html, and the reviewer tried two of these three tubes with the result being more or less gain (12AX7 has the highest amount).
That's right: all 3 tubes are what is called a "dual triode". They all have very high gain & practically the same pin outs. I didn't know that they could be used inter-changeably but then your preamp manuf. might have designed this in.
I think that the 12AX7 might the least microphonic whereas the 12AU7 & 12AT7 are more. My integrated JoLida uses 12AT7 drivers the JJ 12AT7 is quite microphonic. I also think that 12AU7 is also noisier than its peers. I know that my CAT pre Mk1 version used to have 12AU7 but the design now has 6922, which are much quieter.
Sovtek makes a very good 12AX7 called the 12AX7LPS which similar to the Siemens long plate version that was/is considered one of the best 12AX7 ever made. Very popular tube & lots of choices.
The difference is the gain where a 12ax7 = 100, a 5751 is 70 and so on. I often recommend for instance that if you want more headroom choose a 5751 as a replacement for a 12ax7 with a black plate triple mica being some the smoothest and warmest tone. Guitar players who have distortion pedals can substitue a 12au7 due to its lower gain and it makes the overdrive much more usable. If you look online you should be able to find a chart of the difference in gain for all of these family of tubes. Joshua