A display and a printed ticket carry the same words. One is a list the counter has to remember from; the other is an object it can hold.
A screen at the pass is one display showing the orders a counter owes. Staff read from the top, work down it, and clear each line when it is done. A printed ticket is the same order as a single piece of paper, produced at the counter that makes it, with the guest's name across the top. Both tell a bartender or a cook what to make. The difference is that one is a list the counter looks at and the other is a thing the counter picks up, and almost every practical difference follows from that.
It matters most where a venue has several counters and borrowed staff. A screen assumes somebody is watching it and knows which line they are on. Paper assumes nothing - it is lying on the counter, or it is not.
A ticket can be put with what it describes. It sits under the cup, goes in the bag, stays beside a tray that is still being filled. When the order is handed over, the name is read off the paper rather than recalled. A line on a screen never leaves the screen, so the link between the order and the guest standing there lives in somebody's memory for the whole time it takes to make it.
This is the part worth sitting with, because it decides what happens on your worst night. A screen that has frozen, lost its signal or gone to sleep looks identical to a counter with nothing to make. Everything appears calm. Nobody finds out otherwise until a guest comes back to ask about an order the counter never knew existed, and by then there may be a dozen more behind it.
A printer fails visibly. It is out of paper, unplugged or switched off, and it can report that about itself, so the alert reaches a phone instead of waiting to be noticed. A paid order is held until the counter can take it and prints when the counter comes back. The failure becomes a job for somebody with a spare roll rather than a line of guests with nothing to show.
None of this is an argument against screens. A display is the better way to see a whole site at once: which counters are moving, which have gone quiet, what is selling, where the next pair of hands should go. That is a supervisor's view and it is hard to get from paper. The mistake is putting that view where the drinks are made and asking the people making them to use it as their worklist.
So split the two jobs. The counter works from tickets, because its work is one order at a time and it needs something it can hold. Whoever is running the night works from a screen, because their work is the shape of the whole site. There is a quieter benefit too: spiked tickets are a physical count of what each counter made, and laying that beside the recorded sales per station is the shortest honest check a venue can run. A cleared screen leaves nothing behind to count.