"Respectfully" and "respectively" share the root word "respect," which derives from the French "respecter" and the Latin "respectus," meaning "to regard" or "look back at." Both words, then, pertain to looking at or viewing something or someone in a particular way. Both words are adverbs.
When children listen "respectfully" to their teacher while she is giving a lesson, they don't interrupt her or make faces behind her back, because they view her with admiration or authority, which they demonstrate through their actions. The suffix "ful" literally means "full," so these children listen in a manner that fully respects the teacher. Adverbs explain to the reader how something is done; in this case, "respectfully" informs the reader how the children listened.
The adverb "respectively" helps clarify the order of items in a list. For example, in the sentence "Kaitlin and Macy like chocolate and vanilla, respectively," the word "respectively" means that the two pairs -- Kaitlin and Macy, chocolate and vanilla -- match up in order, i.e., Kaitlin likes chocolate and Macy likes vanilla. "Respectively" is usually placed at the end of a sentence to signal the reader to look back at the sentence to glean further understanding.
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