During a recent trip, I saw the gate agent print a long document on tractor-feed paper, delivering it to the plane before pushback. He said it is the passenger manifest.
Why is this printed instead of digital?
Some airlines already have this information in digital form.
If you provide this in digital form, you have to buy $1000 handheld devices every few years, write custom software, have people charge them, pay for data plans, deal with IT security concerns like software updates and unauthorized apps, etc. Compare this against a cost of 2 cents a printed page.
Also consider that the manifest is important in an emergency, it is used to account for passengers in the event of an evacuation, for example. Therefore, you have to satisfy regulators that your handhelds are durable and reliable. And if there is a system failure on the handhelds, you have to delay or cancel the flight, costing thousands of dollars easily. These would be good reasons to simply carry a paper backup.
The linked airline says that their digital app helps improve customer service, by flagging connections and high-tier frequent flyers. That is a good reason, but it may not be necessary for all airlines, such as low-cost carriers.
Beside the points mentioned in the other answer. The manifest helps cabin crew greatly in other things.
In case cabin crew are not sure if a passenger sitting in business class is actually a business class passenger, they would check the manifest to see, it's less hassle than asking for the boarding as some passengers might see that as an insult.
Another reason is knowing passenger names, this is also another service technique used in large airlines. Calling passengers by name without asking them for their names gives a very personal touch with a very positive impression.
Also, the manifest usually indicates the passenger status, is he/she a frequent traveler and what tier.
Lastly, in case of an unruly passenger, most of these guys refuses to hand the boarding or say their names, so it will be easier to get the name from the manifest to fill out required forms before landing.