In this particular case, there is a difference. But only because -- as usual -- the sentence has been modified by a transformation. Twice. By the same transformation.
The noun phrase in question:
- another funny noise, like a mouse being trodden on
consists of the NP another funny noise, modified by a reduced nonrestrictive relative clause, which itself contains a reduced restrictive relative clause.
The original NP would be something like
- another funny noise, [which was] like a mouse [which was] being trodden on
but both of the entirely predictable [which was] parts of these relative clauses have been deleted by Whiz-Deletion, leaving behind only whatever followed the deleted auxiliary forms of be.
That's two tensed forms of be that aren't present (but can be accounted for), plus one being that is present, for which an account is requested.
The first (nonrestrictive) Whiz-deleted which was -- in the clause modifying noise -- contains just the normal auxiliary be required by the predicate adjective phrase like a mouse being trodden on.
It's neither Progressive be nor Passive be, but rather Predicate Adjective be.
The second (restrictive) Whiz-deleted which was -- in the clause modifying mouse -- is in fact the Progressive be, as can be seen by the fact that it's immediately followed by the present active participle (the -ing form) of the next verb in the auxiliary chain (being).
So the third be form -- being -- can't be the Progressive auxiliary. However, since it's followed immediately by the perfect passive participle (the -ed/-en form) of the next and last (and therefore main) verb of the chain (trodden), it must be the Passive auxiliary be.
The auxiliary chain, and Verb Phrases generally, are explained in the VP study guide.
In brief, auxiliaries are not the whole story; the form of the next verb is equally important,
and may be -- as here -- the only structural evidence left after clause reductions.