Alternately means switching between two alternatives, alternatively means doing something different.

> I love pizza and Mexican food, I eat them alternately.

Means Monday I ate pizza, Tuesday tacos, Wednesday pizza again, Thursday burrito, Friday went to the Hospital to get my arteries roto-rooted.

> I love pizza but alternatively I eat Mexican.

Means both choices are good, but says nothing about when or the order in which I eat them.

*Alternately* is about switching. *Alternatively* is about enumerating options.