One factor to consider is timing. If you set up the automatic payments through the bank that holds the mortgage (I'll call them the "receiving" bank), they will typically record the transactions as occurring on the actual dates you've set up the automatic payments to occur on, which generally eliminates e.g. the risk of having late payments. By contrast, setting up auto-pay through your personal bank (the "sending" bank) usually amounts to, on the date you specify, your bank deducts the amount from your account and sends a check to the receiving bank (and many banks actually send this check by mail), which may result in the transaction not being credited to your mortgage until several business days later.
A second consideration (and this may not be as likely to occur on a loan payment as with a utility or service) is the amount of the payment. When you set up your auto-pay through the sending bank, you explicitly instruct your bank as to the amount to send (also, if you don't have enough in your account, your bank may wait to send the bill payment until you do). This can be good if finances are tight, or if you just like having absolute control of the payment. The risk, though, is that if some circumstance increases the amount that you need to pay one month, you'll have to proactively adjust your auto-pay setting before it fires off. Whereas, if you've set the auto-pay up through the receiving bank, they would most likely submit the transaction to your bank for the higher amount automatically.
I'll give an example based on something I saw fairly often when I worked for Dish Network on recovery (customers in early disconnect, the goal being to take a payment and restore service). If you had set up auto-pay through your bank based on your package price, and then the price increased by $2/month, you might not notice at first (your service stays on, and your bill doesn't have any red stamps on it), but the difference will slowly add up until it exceeds a full month's payment, at which point a late fee starts being assessed. From there, it quickly snowballs until the service is turned off. Whereas if you had set that auto-pay up through the provider, when the rate increased, they would simply submit an EFT for the new, higher amount to your bank.
On the opposite side of the spectrum: if you've set up the auto-pay through the sending bank, and you're not paying close enough attention when you finally pay off the mortgage, you might accidentally overpay by either making an extra payment or because the final payment is smaller than the rest. Then you'd have to wait a few days (or weeks?) for the receiving bank to issue a refund, leaving those funds unavailable to you in the interim.
For these reasons, I personally prefer to always set up automatic payments through the receiving bank, rather than the sending bank.