Respectfully means with respect — I respectfully disagree. Respectively means taking each in the order already given: "Anna and Ben scored 8 and 9 respectively" means Anna 8, Ben 9. One is a manner, the other a mapping.
The core difference
A manner and an ordering instruction.
- respectfully — with respect: respectfully submitted.
- respectively — in the order stated: gold, silver and bronze went to France, Japan and Brazil respectively.
How respectively works
It pairs two lists in order, and it only earns its place when there are two lists to pair. "The results were positive, respectively" is meaningless — there is nothing to map onto. If you can delete the word without losing information, delete it.
The sign-off
Respectfully yours is the letter closing. Respectively yours would be nonsense, and it appears more often than you would expect.