They are both Michellin, just different types of tyre. Primacy are "Summer" tyres which is what 90% of drivers use all year round in the UK. CrossClimate are "All Seasons" tyres which are designed for all seasons, so work like Summer in the summer, but perform better than summer tyres in the mud and snow. I have specified "All Season" on my new Cali, as they will be better on the mud of camp sites.
Cross Climate are regarded as one of the best All Season tyres, but they are also one of the most expensive.
On my other car I have just put on the Hankook ones from this review - recommended like CrossClimate, but a bit cheaper.
https://www.tyrereviews.com/Article/2022-Tyre-Reviews-All-Season-Tyre-Test.htm
Worth saying that if you are doing lots of snow driving, then genuine snow tyres will also out perform All seasons in the snow, but will wear out fast in the summer - so people tend to have these on a seperate set of wheels and change with the season.