I'm no black belt but I'll chime in a little bit suing TubeCad as an aid. (See attached)
I am comparing the 12AX7 to the 12AT7 using values provided. I assumed .01 for the coupling cap which affects the low frequency response. I assumed 33k for the input Z which affects high frequency response. I also assumed a Rl=500k which is 5:1 the plate resistor so it is negligible. The AT pulls a little more current so in real world quiescent conditions, B+ should be a tad bit higher with the AX. Gain is about 1dB different per stage. Again not that big of a deal. PSRR (power supply rejection ratio) is about double for the AT so it should be less susceptible to hum. Zout is lower for the AT which in cascading stages is a non issue but it could be a factor if driving a tone stack although probably very slight. Bias and threshold are higher with the AT meaning it can handle more input signal. If you have a chain of stomp boxes in front of the amp, that extra 1/2 volt of headroom could be quite useful. In cascading stages, again that extra 1/2 volt sorta comes into play but more so is the output swing. The AX can make 90ish volts w/o becoming lopsided where as the AT can make 130ish volts w/o becoming lopsided. Of course most of us drive our preamps hard and into the region where one side is clipped. In both cases, the positive cycle clips about 50% before the negative clips. For the most part, both tubes are doing roughly the same thang in these conditions.
Obviously the AT is loafing. I didn't show it but the TubCad report says the AT has insufficient plate current, sometimes called plate starved which sometimes sounds kick donkey in a geetar amp. I run a plate starved EF86 circuit.... but I digress. Look at Imax. The AT can run 10x the current of the AX. RP for the AT is 15k where as the AX is 62.5k. The AT could be happy with a much smaller plate resistor and therefore a smaller cathode resistor to get to a happy bias. In most cases, an AT can sub for an AX with comparable results. This is not necessarily true the other way around. If you have an AT circuit set to idle at 20mA and you put an AX in there, it's gonna fry.
Got a 12AY7 handy? Stick that puppy in there. Under these conditions Vout is almost perfectly symmetrical. Threshold is 2.8V. If you play clean, your brain might melt using this tube.