The 12AX7 has a lot of gain (mu of 100 for those into technical stuff) and so is useful for voltage gain.
However the gain of the tube is traded off with a high impedance and so it is lousy for driving power tubes! Therefore a 12AT7 is employed (one could also use a 12AU7 but it has less gain). You can put a lot more current through a 12AT7 and it has a mu of 70, so you can drive your power tubes and do it with some bandwidth (back when we still used 12AT7 in our driver circuits we got well over 200KHz at full power; we since graduated to the 6SN7 which is a way better sounding tube for this sort of thing).
Phase splitters:
If your amp employs a differential phase splitter (sometimes called a 'long tail pair'; the circuit was made famous in the old Williamson Ultra-Linear design which has been copied endlessly in the world of high end audio despite the design being 60 years old...) the 12AT7 is preferred because its higher mu is desirable in such a circuit. However phase splitting can be done in other ways, such as a 'balanced phase splitter' (which is not really balanced, it just has equal resistances in the plate and cathode circuits). This circuit is very simple and so gets used a lot (the Dynaco ST-70 is a good example). If such a circuit is used quite often the 12AU7 is the tube to use (the Dynaco does not use a 12AU7 but the triode section doing the task has identical properties to the triodes in a 12AU7). Phase splitters are part of the reason that many people prefer SET amplifiers, as a phase splitter often contributes to a 5th harmonic in the amplifier design. For this reason even though we make a push-pull amplifier we don't use dedicated phase splitters.
So Wolf, if you play a guitar this also explains why your preamp sections of your guitar amplifiers are using 12AX7s, but there is usually a 12AU7 associated with the output tubes (an alternative driver tube seen in the old Marshall Plexi was the 12BH7, which is in essence a 6SN7 in a miniature envelope; as such a better sounding tube than a 12AU7...).
By comparison without feedback its hard to get a 12AX7 to go much over about 12KHz. The impedance is too high and stray capacitance is a problem. So 12AX7s are rarely used to drive power tubes because they suck too much at it; instead are used for gain in preamp sections.
