Nothing really
The 12AT7 has a lower amplification factor, but can handle more current.
The 12AX7 has a higher amplification factor, but isn't quite as robust in current handling.
But that aside, the difference is moderate. If you use a 12AX7 you'll give more signal to the output tubes, dont worry about the current use, as they're current-limited, due to the usage, and due to the input impedance of the output tubes. Both of high enough plate impedance that the differences are of little concern.
It might be "wrong" but you can sometimes find a 12AX7 that is mellow, and drives the output better than any 12AT7, but almost never the reverse......
