If you kiss the mob boss's ring, do it respectfully, or full of respect and admiration. But respectively means "in the order given," so if you have to kiss up to the rest of the mob, make sure to shake hands and high five Jimmy Rags and Tommy Two Face, respectively because Jimmy prefers a handshake, but Tommy loves a good high five.
To show respect is to show how you admire something or someone, to show deference or obedience. Therefore, when you do something respectfully, you're being polite because you do it full of respect:
Mr. Ruck listened, as he always listened, respectfully. (Henry James)
"So tonight, we are respectfully asking the candidates to try to put aside the talking points," he says. (Time)
The servants bowed respectfully, and retired in silence. (Bernhard Severin Ingemann)
Respectively, on the other hand, means in a manner that treats several things individually, one by one, in a sequence. It's a way to keep lists parallel, and it rescues sentences from extra words:
By comparison, Mitalas said fellow Metro League schools Seattle Prep and Lakeside had 70 and 40 turn out, respectively. (Seattle Times)
That's easier to say than "Seattle Prep had a turn out of 70 and Lakeside had a turn out of 40." Same goes for this quote from an obituary of actor Cliff Robertson:
His TV performances in "Days of Wine and Roses" and "The Hustler," for example, were filmed with Jack Lemmon and Paul Newman, respectively. (Washington Post)
Sign your emails "respectfully yours" if you're full of respect for the person you're writing to, and save respectively for singling things out. The grammar mob appreciates your attention to these details.