Map<Id, Id> over
Map<String, String> has at least two benefits I can think of.
The more restricted type identifies that this isn't just a map of string-string key-value pairs, it is specifically a map of id-id key-value pairs. An experienced programmer, in particular, will immediately understand that we're making a link between records via their ID values. The string-string version leaves ambiguity if used improperly, so one should prefer the id-id version when practical (and, alternatively,
Map<Id, String> and
Map<String, Id>, if that's the intended use case).
Sometimes we have to deal with 15-character and 18-character Id values. Mix them up in a map with a string key and you'll introduce logic bugs, since 001000000012345 is not the same as 001000000012345AAA. Using the correct key removes the possibility of a false negative when checking if a key is present or getting the associated value.
In summary, using the correct data type helps make the code self-documenting and introduces clarity in to the code, especially for the next person who's unfamiliar with the code (possibly including you a year from now). In some edge cases, it can also prevent false negatives by coercing the data into the correct format, either by the compiler or at runtime, and logic bugs can be easily identified when the correct type is used (e.g. if you try to put a non-ID into an Id key, you'll get an exception).
In any case, there is no benefit for using
Map<String, String> over
Map<Id, Id> when the situation calls for using Id values. Instead, it only risks the possibility of logic bugs and confusion.