Both of those particles can mean "too", "even" or "any" (when used with wh-question). Do they have any difference in meaning or connotation when used like this? E.g.
も and だって are different.
だって is an informal version of でも and is used in the same way, i.e. Noun+でも = even (the noun). も merely expresses 'too' or 'also' when combined with a noun. So the difference is as follows:
私もフランスに行きたい。 I also want to go to France.
私だってフランスに行きたい。Even I (would) want to go to France.
だって doesn't mean "too" or "also" but only "even" or "any".
So, you can't say あなただって行きたいですか？ for "Do you want to go too?" or reply as 私だって行きたい to "Do you want to go together?" though you can say あなただって行きたいでしょう？ for "You would like to go there even if you were in the position, wouldn't you?".
You might want translate 私だって行きたい as "I want to go too" but more precisely it's "even if I were in your position, I'd like to go".