Here's an example:
Demonstrating that the assumption $A=B$ leads to a true statement is a vacuous truth.
In order the show that $A=B$, prove that the difference $\Delta =A-B$ is zero. The subtle change being that $\Delta$ is not assumed to be zero.
What are some other examples of subtle logical pitfalls that the amateur Mathematician should be aware of?
Here is an specific argument that shows how assuming $A=B$ leads to absurdity.
$$ \begin{eqnarray} 2&=&1\\ 2-1&=&1-1\\ 1&=&0\\ \end{eqnarray} $$
$$ \begin{eqnarray} a+b&=&a+b\\ a+1*b&=&1*a+b\\ a+0*b&=&0*a+b\\ a&=&b \end{eqnarray} $$
A falsity implies anything. Assuming that the false statement is true implies that the two undefined objects $a$ and $b$ are equal, absurd. However, if we define the difference as $\Delta$, then a true statement is forced.
$$ \begin{eqnarray} 2-1&=&\Delta\\ \Delta &=&1\\ 2&=&\Delta +1\\ 2&=&2 \end{eqnarray} $$