These two words look almost alike but the meanings are quite different.
ALTERNATELY means to occur in turns, repeatedly: “Jessie and Samantha alternately cleaned their cat’s litter box. Jessie did it in the morning and Samantha’s turns were at night.”
ALTERNATIVELY offers another option: “Jessie, you can clean the litter box every morning or, alternatively, you can do it morning and night for a week and then Samantha will clean it the next week.”
Be sure to proofread everything you write; your spellchecker will not highlight an incorrect word if it is spelled correctly. You are smarter than your software.