The final hash verification is the same one done in mining. The difference it's that is done just once, while in mining it's done lots of times, until the miner finds a hash on the target difficult.

It's similar to checking a database of pictures of stores to tell if there's one that's a pizzeria. You look to hundreds, even thousands of random pictures, your eyes and brain working on what you see to tell what the picture is, until you find a pizzeria. Then you post it online and people see it and they say, 'yeah, it's a pizzeria'. They only had to look at a single picture to validate it.