For hard forks, after the change in Bitcoin protocol, old nodes will reject blocks created by new nodes since it does not conform to the rules. Miners will follow the original rule, creating a new block on the latest verified block.
For soft forks, old nodes won’t detect the change in Bitcoin protocol and continue to accept the blocks created by new nodes. Miners may work on blocks that they didn’t fully understand or validate.
Soft forks and hard forks are backwards compatible, meaning that new nodes would be able to verify the blockchain from scratch. Backward compatibility refers to new software accepting the code produced by older software.
For example, Windows 10 can run Window XP applications. Soft forks are forward compatible, referring to older software accepting the code produced by new software.
A document you saved using Word 2013 can be opened with Word 2011. That is an example of forward compatibility.