You made a logical error.
You started with an equation that you had no idea if it had any solution. Then you made a series of transformations and found out that the only possible solution (in the real numbers) could be $a=b$. Then you put that into the original equation and found that $a=b$ leads to something that is untrue. Alternatively, the problem statement removed the possibility of $a=b$ outright.
So the only logical conclusion is that this equation has no real solution!
You can easily see that this is actually true, because at some time in between you arrived at
Multiplying that by $2$ leads to
This can only be true if $a=b=0$, which is impossible because the orignal problem statement had $a$ and $a+b$ as denominators!
The core of Andrei's answer is that the transformations you used to modify the equation step by step do not always produce equations with the exact same set of solutions. That's already true for the first step, which yieled
It certainly has the solution $a=b=0$, which isn't a solution of the equation you started with.
The reason is that some kinds of transformations, like multiplying both sides of an equation with a number, or squaring both sides of an equation, are not equivalent transformations. That means the transformed equation has all the solutions of the untransformed, but it may have more.
So in order to check if what you found as solution of your many times transformed equation is actually a solution of the original equation, you have to substitute the found solution into the original equation.
In this case, it lead to the result that non of the found solutions is actually a solution of the orignal equations.