师父 vs 师傅

What's the difference between 师父 and 师傅, other than how it's written?

师父
1) can also mean 师傅
2) a honorific title for monks, nuns, and priests

师傅
1) the teachers who impart their profession skills to others
2) a honorific title for people who are skilled in their profession, eg 厨师傅(chef) 木匠师傅(carpenter)

In olden times, the people/disciples who learn from 师傅s usually start learning when they were very young. The 师傅s' ages thus were around the same as the disciples' fathers. Added to that was the fact that the disciples treated the 师傅s with respect akin to the respect they gave their own fathers, plus usually the 师傅s were male, which often resulted in 师傅 being written as 师父, and people in general accepted this.

Nowadays, though, 师傅s are not necessarily someone as old as the student's father, and might also be a female, so to address a teacher, using 师傅 would be more appropriate.

On the other hand, using 师父 to address a teacher would sound more affectionate, as there is an underlying meaning of 师父=父.

Can one use 师母 instead of 师父 to address a female teacher then?

In this day and age, the term has evolved to include female teachers, though it is more commonly understood that 师母 == 师父's wife.