Assume seat selection has already been done, the flight is an international one so the airline may want to see documents, and the passenger is going to check a suitcase.
Is there any advantage to checking in on-line and printing a boarding pass?
Well, it'll depend, but yes, there are certainly advantages.
Often airlines have bag drop only, if you've checked in online. They'll still check your ID, but it saves you lining up with all the others who still need to go through the check in process.
By checking in early, you're less likely to be bumped - flights oversell regularly, on the assumption a certain percentage won't make it. As a result, if you really want to make sure you get on the flight, you check in ASAP.
If you've got status, it gives you more time in the lounges, to eat, drink and relax. It also gives you time to shop, if that floats your boat.
On some carriers (less these days as many charge), by checking in early you have the opportunity to ask nicely for an exit-row seat - usually you can't check in to these online as they have to make sure you're fit and able to comply with the crew in an emergency, so they give them to people in person at bag-drop/checkin.
Some airlines use checkin time as a tie-breaker for upgrades. Generally these days they're stingy and only give it to people with status, but all things being equal, if you checked in first, you may get it. But perhaps the most important - for international flights - things go wrong. Your taxi doesn't turn up, there's a failure on the train line, you forget something at home and have to go back - all of these have happened to me. By doing things ahead of time, it's one less moment of stress at the airport, and gives you that extra time you might need. And importantly, if check in has closed when you get there, it's too late to check in, but if you've checked in ahead of time, you have a boarding pass in your hand and can still go through security and try to make your flight.
Some airlines, like Southwest, have a different seating policy according to checkin. Since the seating is modeled as a "sit wherever you like, as long as there is space", you can sit anywhere you want, but your choice really depends on how early in the queue you get.
If you check-in early, obviously you will be placed at an earlier boarding position, and thus have more choice as to the seat you want, whereas if you check-in late or a few hours before the flight, you will most likely be restricted to sitting in the back or between people (due to the "good" seating being taken up)
Depends in part on where you are traveling from, but in general No.
In the US, boarding passes need notation that documents have been checked. So even if you have printed boarding passes at home, the check in counter person will likely print out new ones with your document status noted. If just a single flight they may hand write on your home printed pass.
As more and more people check in online, the lines at the bag drop counter are getting longer and at the check in counters getting shorter. And for international travel the bag drop person performs almost the same computer entries as the check in person, so no real time saved at the counter.
Occasionally check in time figures into decision making process for both involuntary bumps and upgrades. But it is, as Mark point out in his upgrade mention, primarily used as a tie breaker.
On the plus side would be peace of mind, as you would have confirmed that you got your seat of choice. Have a heads up on any situations you might face the next day, since airlines often ask for volunteers at check in if they are facing an oversold situation. Maybe an earlier opportunity to purchase a gate upgrade.
Depending on airline, you can also do baggage drop-off the day before once you are checked in. So if you don't want to leave your car at the airport, you bring your suitcase by car (cheap short-term parking near the terminal), and then use public transport on the next day to get to the airport.
Other than that, not much difference.
The availability of space in the overhead bins for carry-on luggage is a pretty big one on some airlines. Many airlines determine boarding order at least in part due to time of check-in. This is especially true on Southwest Airlines where you are assigned an actual position in the boarding queue based on time of check-in (and a couple of other factors.) However, it is also true to a lesser extent on some of the other airlines (such as Delta) where those who check-in earlier tend to be in earlier boarding zones than those who check-in later (though still behind everyone who has 'priority' boarding, which seems to be about half the plane in the case of Delta.)
Since overhead bin space in economy is typically first-come, first-serve, if you check in late and, thus, board late, all of the overhead bin space may be used before you get onboard, resulting it your bag being checked and placed in the cargo hold instead of riding with you in the cabin. If you didn't previously have any checked baggage, this means that you'll now have to wait around for your bag once you arrive at your destination airport. It can also mean being separated from fragile/valuable/etc. things that you'd rather not have ride in the cargo hold as well as the possibility of it being delayed or lost, as happens sometimes with checked baggage.
Of course, as another answer has mentioned, in the case of airlines like Southwest where seating is also first-come, first-serve, checking in late also frequently means getting stuck in a middle seat, and, if traveling with someone else, your party being separated. Even some airlines that do assign seating don't do so until check-in, so waiting until later to check in will also frequently mean not getting the seats you want on those airlines, too. This was the case with an Air France medium-haul flight that I was on recently, for example, though I think they assign seats at booking for longer flights. If I remember correctly, this was also the case with Swiss, except that they gave you the option of paying a small fee to select your seat at booking instead.
There is a major advantage particularly if you have a complicated air ticket. Completing the check in and obtaining a boarding pass proves (in almost all circumstances) that your eticket is valid for the flight segment that the boarding pass covers.
If you are making a lot of changes to your ticket, or if you have many different flights operated by different airlines on the same ticket, the ticket can actually become out of sync with the reservations and the PNR, and this is unfortunately quite hard to detect. Around the world tickets are particularly bad for this because of the degree of manual intervention required.
When you try to check in, this is a good test that the ticket is valid, although online check in can fail for many other reasons. Checking in early gives you more time to resolve any problems with the ticket.
If you're late arriving at the airport, the check-in desk and self-check-in kiosks for many airlines will not let you check in, even if you still have enough time to make it through security and board. If you're already checked-in online, they have no way of knowing what time you actually arrived at the airport, and thus no way of enforcing arrival time cutoffs.
There can be an adventage in checking early.
This is pure gamble tho.
An advantage not yet mentioned is that it may provide one advance notice if the flight is going to be delayed or canceled. While it would be good to speak with an actual human to find out the earliest time before which the flight could not possibly leave, being able to spend extra time with friends or relatives at a restaurant near the airport rather than spending it alone at the gate may be desirable.
Further, if an airline can't carry a passenger at the planned time, it may sometimes be more than happy to offer a passenger a choice of alternative flights. While airlines will by default try to get passengers to their destination as soon as possible, it may sometimes be better for all concerned to have the passenger travel on the next day. The sooner one can find out about a delay or cancellation, the more effectively one will be able to exploit any opportunities it might present.
Many airlines charge extra for well-located seats, even in economy. Most of the time, if there are any well-located seats left, there is no extra charge to change your seat to them during check in. Even if I have a clerk check me in at the airport, I always ask if there are better seats, unless I'm late.
My mom tried to check in guests that didn't really know how to use a computer. It turns out that since they booked the flight 6 months ago, the flight date had changed, and instead of trying to check in 24h in advance, they had to rush to the airport just to be on time. Unless you have a good reason, always check in early.
Yes! The airline may fly you out earlier! On a different and earlier flight! Then, you can get to where you are going EARLIER. Done deal. Bam.