Email is like a 100,000 times faster than snail mail. Proof of work is like 0-1,000,000 times more expensive due to duplicating work than a database.
For the end user sending a transaction is quicker and cheaper than getting your lawyers to draft contracts, maintaining your own books, paying an accountant or someone else to manage/balance the books, coordinating a deal/terms with them, etc.
Also there's Proof of Stake which is much more efficient than Proof of Work.