Question: why simplify works on A-B giving zero, but not on A=B ?

I was trying to simplify long expression of the form    f(x)=g(x), where f(x) should simplify to be the same as g(x) and get 0=0. But Maple simplify just return the input back.

Then I wrote simplify(f(x)-g(x)) and now Maple returned 0 back.

What is the difference between simplify (f(x)=g(x)) and simplify(f(x)-g(x))? And why the second worked but not the first?

restart;
f:=(9*(x^(-2/3*a))^2*exp(6/a*(x^(-2/3*a))^(1/2))^2*_C0^2-6*(x^(-2/3*a))^(3/2)*exp(
6/a*(x^(-2/3*a))^(1/2))^2*_C0^2*a+x^(-2/3*a)*exp(6/a*(x^(-2/3*a))^(1/2))^2*_C0^
2*a^2+18*(x^(-2/3*a))^2*exp(6/a*(x^(-2/3*a))^(1/2))*_C0-2*x^(-2/3*a)*exp(6/a*(x
^(-2/3*a))^(1/2))*_C0*a^2+6*(x^(-2/3*a))^(3/2)*a+x^(-2/3*a)*a^2+9*(x^(-2/3*a))^
2)/(3*_C0*exp(6/a*(x^(-2/3*a))^(1/2))*(x^(-2/3*a))^(1/2)-exp(6/a*(x^(-2/3*a))^(
1/2))*_C0*a+3*(x^(-2/3*a))^(1/2)+a)^2:

g:=x^(-2/3*a):

And now

simplify(f=g);

But

simplify(f-g);

                                                           0

And

simplify(f-g=0);

                 0 = 0 

Why Maple behaves like this? I did not know it makes difference if one writes f=g vs. f-g in terms of simplification. 

I copied the code to Mathematica to see how it behaves, and Mathematica Simplify worked on both f=g and also on f-g as one would have expected:

 

Maple 2020.2 on windows

 

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