Are there any downsides to paying rent this way?
No. Where I live there can be upsides. Some banks charge for paper transactions involving checks (cheques).
Is that different, legally, from "wire transfers"?
Yes, wire transfers are effectively anonymous and can be picked up as cash without a bank account, which is why they are popular with crooks. Electronic payments go into a known bank account and are traceable. The bank should be able to provide proof of payment in any dispute.
Note: the use of these terms vary from country to country. "Wire transfer" may mean something different in your locale.
What if the payee (or the payer) makes a mistake in the account number,
You'd have to ask your bank if the erroneous payment can be withdrawn.
My bank's online banking service allows you to set up account details for people you expect to make future payments to. This reduces the chances of making a mistake as you pick the recipient from a short list rather than typing in a number.
or the payee's account runs into problems?
If the payer's account has insufficient funds, either the bank will refuse payment or your account will be overdrawn (and you will be charged heavily for if no overdraft facility has been arranged in advance). I think this depends on the bank. I've seen both situations.
I don't know what happens if the payee's account is locked or somehow unable to accept the payment. I'd be astonished if the money disappeared from my account, didn't arrive in the payee's account and the bank could give no satisfactory explanation. I'd expect the money to remain in the payer's account.
Example Western Union money transfer option
See also "Wire transfer scams" "Crooks like their victims to use wire transfers because the money moves fast and they can take the money and run before the victim discovers the truth,"
So yes "wire transfers" are different from bank-account to bank-account online transfers.