What a nuance!
alternate
We decided to make alternate arrangements in case the hotel was fully booked.
We decided to make alternative arrangements in case the hotel was fully booked.
We decided to make alternative arrangements in case the hotel was fully booked.
In British English alternate and alternative have different meanings.
alternate
= (1) happening in turn, first one then the other: 'alternate periods of sun and rain'
(2) every second (day, week, etc): 'Our local football team plays at home on alternate Saturdays.'
alternative = that may be used instead of the usual one or the one you had planned to use:
'In view of the roadworks, motorists are advised to use an alternative route.'
In American English alternate is used with the same meaning as alternative.
P.S. When you pick an alternate route, you have substituted one route for another. If you took an alternative route, you have chosen to travel on that route rather than another.
Alternate - one in a series, a substitute for something, occurring or succeeding in turns
Alternative - offering or expressing a choice, something that can be chosen
The difference is choice. There is no choice with an alternate. You have a choice with an alternative.
Here's a heated argument about the word choice.
