Lets do the easy one first. TrueQ[x^2 == x*x] returns True since x*x is converted to x^2 automatically before the test is even done. If you type x*x you'll see the FE returns x^2. Hence now both the left and right side are exactly the same When the kernel gets hold of them. a==a which is Equal[a,a] automatically converted to True for any a. So it becomes TrueQ[True] which returns True

For the other examples, the argument to TrueQ was is not explicitly True, hence TrueQ returned False. The argument to TrueQ has to be an explicit True for TrueQ to return True.

You need to use SameQ for the other 2 examples

SameQ[(m^2 - n^2)^2 + (2 m*n)^2 == (m^2 + n^2)^2] gives True. This is because SameQ[Equal[a,b]] return True if Equal[a,b] itself returns True, which it does in this case. Simplify@Equal[(m^2 - n^2)^2 + (2 m*n)^2, (m^2 + n^2)^2] gives True. So SameQ[True] is True

The bottom line is this: TrueQ[expression] is the same as SameQ[Equal[expression]]