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.
