What happens if its summer and I want to book my winter break or winter and I want to book my summer break. I would make sure that they are definitely sure they want two versions of the site before I started but if they are then there is a possible 3rd option to consider - subfolders - www.example.com/summer and www.example.com/winter
For 6 months you redirect anyone coming to the www.example.com site to the required seasonal version. Any links in will to an extent benefit the whole site rather than the one subdomain of the site.
One problem you are going to face is that if you build up links to the summer domain or folder then switch to the winter site some people may remove their link to you as it no longer points to the page that it did when they set up the link.
My preferred option would be a home page giving the option to get to navigate to both sets of content which is held in subfolders with the focus of the home page (in terms of content and design) switching when the seasons change.There would be no need for redirects, less SEO issues, better for a user who is planning ahead and means that any links achieved by either section of the site benefit the domain as a whole.
If wanted to minimise the footprint of the link to the other seasons site then like some sites that have small country selectors at the top of the screen there could be a small season selector that a user only uses when they want to view the off season site.