Alternate (ly) and alternative (ly)

Alternately means 'every second', 'first one and then the other', 'in turns'.

We spend alternate weekends at our country cottage.

I'm alternately happy and depressed.

Alternative (ly) is similar to 'different', 'instead', 'on the other hand'.

Janet's not free on the 27th. We'll have to find an alternative date for the meeting.

You could go by air, or alternatively you could drive there.