'Good with' means good at using or dealing with something. Common collocations are 'good with your hands' (for someone who can do woodwork, for example) or 'good with people/children/animals'. If someone says 'I'm not good with accents', this suggests that the person can't cope well when they hear accents and has trouble distinguishing between them. This is different from an actor who is 'good at accents', which means that he or she is skilled at performing in different accents. Likewise, if a person says 'I'm not good at presentations', this means that they are not skilled at giving presentations. Compare this to 'good at sport' 'good at music' good at languages' - these all describe skill.

Good question! "Good at" is definitely the more common phrase. You can be goods at sports, academic subjects, or various activities. "Good at (doing) presentations" would fall in the last category. "Good with" is used less frequently, is usually followed by a plural noun and isn't necessarily interchangeable with "good at". "Good with children" wouldn't be right if you use "at", although "good at numbers" is possible, "good with numbers" sounds more natural to me. your example of "good with accents" would also be acceptable as "good at accents".