How to eliminate == sign from the solution?

a + b == ab - c + d is stored internally as Equal[a + b, ab - c + d].

And what you want instead is Subtract[a + b, ab - c + d].

You can get there by replacing Equal with Subtract by doing this:

f = Apply[Subtract, a + b == ab - c + d]

which gives you a-ab+b+c-d, but that is the same as what you asked for, just with Mathematica sorting the variables into the order it wants.

one way might be to make a lhs() and rhs() functions.

ClearAll[a, b, c, d, x, y, lhs, rhs]    
lhs[eq_] := eq /. (x_) == (y_) -> x;
rhs[eq_] := eq /. (x_) == (y_) -> y;

Now you can write

eq = a + b == a b - c + d;
f = lhs[eq] - rhs[eq]

This will work only for input of form x==y of course.