Let $L_1$ be a regular language and $L_2$ any language. I want to prove that the language of $L_1$ truncated by $L_2$ is a regular language, i.e.
$$\{w| wx\in L_1 \text{ and } x\in L_2\}$$
is regular.
To prove that something is regular it is enough to prove that it is described by a regular expression. We know that $L_1$ is described by a regular expression, but regular expressions don't handle "differences" in any obvious way.
We could imagine the DFA which accepts exactly the strings of $L_1$. But to truncate the strings of $L_2$ we would need some way of recognizing the portions of the tail of the word which are in $L_2$ and there may be no DFA which does this. Likewise for any recognizing DFA.
In general it's hard to imagine having "any language" interact with a regular language to produce a regular language. The "any language" isn't constrained by any of the tools that seem relevant to regular languages, like NFAs and regular expressions.
