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No, there are no such examples known. In fact, with the current technology, the two questions are more or less equally hard. That's because for any given prime $p$, you can, in principle, establish finiteness of $Ш(E/\mathbb{Q})[p^\infty]$ algorithmically by performing $p^n$-descent for higher and higher $n$, until the upper bound on the rank of $Ш(E/\mathbb{Q})[p^n]$ stabilises. Of course, we cannot prove a priori that this would ever happen, but in practice, if you knew finiteness of $p$-primary parts of sha outside a finite set of primes, you would run your computer to do $p^n$-descent for the remaining primes, until you establish finiteness for this finite set, too.