There’s no difference

It is all about the concept of register in languages. Three times a week (informal)+spoken Three times per week (formal) written or scientific

They both mean the same thing. 'A' can be found defined in the dictionary as 'per'. The only suggested difference is 'a' is more conversational whereas 'per' is more formal or careful.

Both phrases are correct. Per is more common for expressing quantities such as 60 mph 60 miles per hour.

I'd say that "three times per week" sounds a bit more formal and precise. "Three times a week" sounds a bit more conversational and in some cases there might be room for imprecision, although if it's only three times, a week, it's likely that that's an accurate number.