An Alphabet containing only $A$ and $B$.
Removing $AB$ makes no difference to a word and adding $BA$ or $AABB$ makes no difference to a word.
Is $ABB = BAA $?
ABB can only be equal to words that have one more B than A. So BAA is not achievable.
Observe that if two words have no difference, then the difference of the number of letter $A$ in the word and the number of letter $B$ must be same. In the problem, $ABB$ has the difference of -1 but $BAA$ is 1. Therefore, $ABB$ is not equal to $BAA$.
Am I Correct?
$ABA=A$ (By removing $AB$)
Now by adding $BA$ will not make any difference, so
By removing $AB$ we get $BAA=A $(1)
Clearly , $ABB=B$ (2)(By removing $AB$)
Now by (1) and (2), we have $BAA=A$ and $ABB=B$.
Since $A$ and $B$ are unique, $BAA \neq ABB$