Εντελώς and τελείως mean exactly the same thing so completely, entirely, totally, absolutely, utterly, etc. all could work fine for both words. If you use perfectly as completely, it also works for both, but if you mean flawlessly then neither is a correct translation, you should either say τέλεια or άψογα. For example "I'm perfectly healthy" - Είμαι εντελώς/τελείως/απόλυτα υγιής but "I answered the questions perfectly" - Απάντησα στις ερωτήσεις τέλεια/άψογα.
I don't think that's ever the case and can't think of any examples where that would be possible. Wordreference does not offer any examples either, so it seems it's a no. Wordreference.com is a very good resource for translations in general and usage in particular when there is meaning overlap. Please check the two link below for usage of the two terms in the two langauges: