Chalk and cheese
Definition 1 of 1
Pro Tip 1/3British Origin
Used mostly in British English, less common in other English-speaking regions. 
In American English, they might say 'apples and oranges' where we say 'chalk and cheese.'
Pro Tip 2/3Usually Negative
Often points out incompatibility or a striking difference that's not viewed positively. 
Trying to pair those two together was futile; they're like chalk and cheese.
Pro Tip 3/3Strong Contrast
Emphasizes extreme differences, more than just a simple contrast. 
It's not just that their opinions differ - their entire worldviews are chalk and cheese.