Yeah, that's more or less true, but gain is very often irrelevant as the issue is rarely solely about gain. Many times the required gain is low enough, typically about 50 in most circuits, that either tube could be used if the issue was purely about gain, and, again, it rarely is this simple. The maximum gain is an obvious difference between the tubes, true, but that often little matters in actual circuits where we have more than enough gain to go around.
As a general rule, the 12AX7 is generally used a high-gain voltage amplifier while the 12AT7 and 12AU7 are generally used as high-gain current amplifiers. The input and output impedance are different, as is the transconductance, and the plate current. The tubes, in short, are different.
A current-amplifier is often as a unity-gain voltage device (ok, it isn't quite unitary, but this is written for pedagogical purposes) to improve current drive. This is why tubes are often seen with grounded grids where current gain is the desired effect.
Just because the 12AX7 has a particular reputation among those who don't understand tube circuitry does not meant that dropping it in willy-nilly improves the circuit. Usually quite the opposite.
Swapping tubes of different types is generally doomed to fail because the tube characteristics are never quite the same and the circuit, including passive component values, was designed for a particular device's properties. Unless you understand the circuit at least as well as the original designer it is dubious that swapping tubes will improve the sound. The same goes for opamps and transistors. Substitutions randomly made without understanding the ramifications can result (at best) a degradation in performance and (at worse) oscillation or damage to the equipment.
It's not about the size of your gain, it's about what you do with it.