400 Hz transformers are smaller for the same kVA. If you operate a 400Hz transformer at 60Hz but with the same supply voltage, then you will push the core into hard saturation and excessive current will flow, letting out the magic smoke.
When a transformer is operating, current in the coils generates magnetic flux in the core. Since you have _changing_ current (AC) you have _changing_ flux, and this changing flux induces voltage in the coils. This voltage opposes the applied terminal voltage, and is what limits the current flow. Once the flux reaches the point that the magnetic core gets saturated, then the changing current causes much less change in flux, and thus much less opposing voltage...the result being much greater current flow.
The key number is the 'volt-second' product, how many volts are applied for how much time. The greater the volt-second product the more core you need to carry flux.
At 400 Hz the cycles are shorter, so for the same voltage the volt-second product is smaller, and so you can get a way with a smaller magnetic core, or fewer turns of wire. This makes the transformer smaller and lighter. The same applies to motors, and the smaller hardware is the reason that 400 Hz is used in aircraft.
So you can use a 400 Hz transformer at 60Hz, but only if you reduce the voltage by 6/40. You get fewer kVA out of the same mass of copper and iron.
There are secondary differences, for example you have more eddy current losses and thus need thinner laminations in the 400 Hz transformer, and other small details.
-Jon
