Alternately means several things taken in turn, consecutively, one after another. Alternately is an adverb formed by adding -ly to the word alternate, which comes from the Latin word alternatus meaning one after the other.
Alternatively means another choice, on the other hand, another possibility. Alternatively is an adverb formed by adding -ly to the word alternative, which also derives from the Latin word alternatus. Remember, alternately refers to several things following each other in turn, alternatively refers to a different, separate option or choice.
Maybe the Vietnam War offers some perspective: from 1969 on, the US and North Vietnamese alternately escalated and eased military activity while peace negotiations dragged on and off; the US was especially active in launching massive bomb campaigns to encourage North Vietnam to accept an accord. (The World Post)
“The Things We Keep” is set alternately in 1968 and 2003, following the same characters in both time periods. (The Chicago Tribune)
Dreyfuss is plenty convincing as the alternately cool and extremely high-strung crook. (The News & Observer)
Fold in the flour mixture alternately with the buttermilk in three additions, ending with the flour and scraping well in between. (The St. Augustine Record)
The second approach professes that markets are too efficient to allow for “better” indices, and that outperformance can be best obtained by exposure to these very same investment factors — which is not something the makers of alternatively weighted products care to admit. (The Financial Times)
Alternatively, they might have a sense of the problems with treats but find it hard to change habits or to stand up to your children and make the necessary changes in the routines. (The Irish Times)