It seems like the creators tried to stop the various tool proficiencies from overlapping with skills, but one area where a redundancy does exist is with instrument proficiencies and the perform skill. The perform skill seems to make you proficient with all instruments and thus eliminates the need for instrument proficiency.
Oddly, this means the bard, the class that should be the most likely to take perform, really doesn't need it since they gain three instrument proficiencies automatically which would be rendered redundant if they choose the perform skill (now admittedly, a bard who wants to perform in ways other than instruments could use perform, but it's going to be very seldom when the ability to sing or perform puppetry will solve a problem that playing an instrument could not.)
I almost feel like there should be an exception and you can add double your proficiency bonus if you both have the perform skill and are proficient with the musical instrument in question.
Oddly, this means the bard, the class that should be the most likely to take perform, really doesn't need it since they gain three instrument proficiencies automatically which would be rendered redundant if they choose the perform skill (now admittedly, a bard who wants to perform in ways other than instruments could use perform, but it's going to be very seldom when the ability to sing or perform puppetry will solve a problem that playing an instrument could not.)
I almost feel like there should be an exception and you can add double your proficiency bonus if you both have the perform skill and are proficient with the musical instrument in question.