Most asian motors are massivly over rated as far as H.P. goes, so are a lot of hand held power tools. There are lots of 3 hp routers forsale that have 120 volt 15 amp cord ends.
In the case of your lathe, it will have both start and run capacitors to help the motor simulate a 3hp motor for short time spans. The capacitors or the motor itself will burn out if subjected to such loads, asuuming you don't have a proper motor starter equipped with overload heaters.
Bill there should not be any difference in performance in your lathe, you are still putting in the same amount of energy to the motor(voltamps). WHen you change the motor connections from 120 to 240 single phase, and apply 240 volts power, the current (amps) drops to half of what it was on 120 volts.
A single phase capacitor start/run motor rated at 3 hp will not run a constant 3 hp load, unless it is rated for continuose duty. All asian motors are vastly overated, even there 3 phase ones.
A continuose duty, 3 hp single phase, that will actually put out 3hp, will draw around 21 amps on 120 volts at full load. 240 volts the current will be half of that ,or around 11 amps. The amperage draw at start up will be much higher for each.
These days with relativly cheap frequency drives available that will also create 3 phase power output out of single phase power input, it is much better to equip your tools with 3 phase motors.But this is a whole different subject.