Paper card or online card, what is the difference

There is no single answer. It depends on who is writing, where, and how many. Written honestly by someone who has run both.

What paper does well

The warmth of handwriting. The letterforms, the pressure marks, the crossed out and rewritten words all carry the person. Ten fonts cannot imitate that. Even bad handwriting is welcome because it is theirs.
It becomes an object. Finding a card from ten years ago while cleaning a drawer only happens with paper. A file must be searched for; paper catches your eye by accident.
Writing becomes a ritual. Sitting down with a pen, thinking of one person, has meaning apart from what gets written. The writer remembers that time too.
There is a moment of handing it over. Giving, unfolding, reading. It weighs more than sending a link.

Passing it around takes time. One person writes and hands it to the next, so ten people take days and thirty take weeks. If someone puts it in a bag and forgets, it stops there.
One person losing it is the end. If the twenty-first person leaves the sheet twenty people wrote on a bus, there is no recovery. It happens more often than you would think.
People far away cannot write. The friend studying abroad, the colleague transferred to another city, the family out in the provinces are left out. Mailing it around is not realistic.
Nothing can be fixed. Typos and sentences written in the heat of the moment stay. When space runs out, people write over the previous message.
Taking turns is pressure. When it is your turn you must write right there. Messages written without time to think usually end in “congrats, take care”.

No limits of distance or time. With one link, someone overseas writes too. Everyone comes in when it suits them, thinks, and revises. Participation is clearly higher than on paper.
Everyone writes at once. No waiting for a turn, so even thirty people are done in a day. The board updates itself, so you can see how many notes have come in.
You can fix your own message. Spot a typo after posting, or think of a better sentence, and you can reopen it on the same device and edit. You cannot touch other people's notes, so there is no risk of ruining them by accident.
It survives as an image. Save the whole thing as one PNG with the image export and it cannot be lost, and everyone who took part can keep a copy.
It costs nothing. No sign-up, no app, no payment. Not even paper and a pen. How to make one is here.

Less of the handwritten feel. Switching to a handwriting font does not make it their handwriting. On desktop you can draw with the pen, but mouse writing is not pen on paper. Honestly, paper wins here.
The link has to be managed. Anyone with the link can read and write. For a surprise you must make sure it never lands in a chat the guest of honor is in, and a lost link is hard to find again.
It depends on a device. You need a phone or computer to write and read. Send an elderly relative a bare link and they may never open it. There are ads on screen too.
The handover is weaker. If it ends with sending a link, the recipient may scroll once and move on. The delivery needs its own thought.
Permanent storage is not guaranteed. A free service cannot promise to be there forever. Saving the finished card as an image is a given.

Same classroom or office, ten people or fewer: paper. Passing it around takes a day and you keep all the warmth of handwriting.
Twenty or more: online. Twenty people on paper means missed deadlines and a stalled sheet. With a big group, the list layout reads best.
Even one person far away: online. That one person is often the one who most wants to write.
A surprise: both work, but online is less likely to be found out. Paper gets discovered lying on a desk. You do have to check who gets the link.
For an elderly recipient: the final handover is better on paper. Collect online if you like, but print it and put it in their hands.
Less than three days left: online. There is physically no time to pass paper around.
Formal occasions like farewells and retirements: either, but collecting online and printing it for a frame gets you the formality of paper and the participation of online at once.

The highest satisfaction actually comes from mixing. Take the strengths of online for collecting and the strengths of paper for the handover.

Collect online, print, hand over. Gather messages through the link for a few days, save the image, then print it as a photo or on A4 and give it like a card. Messages from far away end up on paper, and there is still a moment of handing it over. Details in Print and frame.
Photograph a handwritten note and add it as a sticker. Anyone who really wants their handwriting in it writes on paper, photographs it, and uploads the photo with the sticker tool on the canvas. Any image under 5MB works as a sticker, so a photo of a handwritten letter sits on the canvas next to the sticky notes. Uploading straight from a phone is faster than moving the photo to a desktop.
Paper at the party, online for those who could not come. Pass a paper card around at the farewell dinner, collect the absent people's messages online, print them and tuck them into the card.
Paper first, online later. Hand over a paper card on the day for the moment, then send the online card image a few days later for a second wave.

Going online does not mean dropping the habits of paper. Setting a deadline, the organizer writing first, and creating a moment of handover are things learned from paper cards, and they work just as well online.
When uploading a photo of a handwritten note, check that the writing is legible. A small photo gets smaller on the canvas. Drag a corner to enlarge it after placing, or crop to just the writing before uploading.
Remember that online, anyone with the link can see it, and that paper is a single copy that is gone if lost. Whichever you choose has a weak point.
If you made it online, save the image right after it is finished. Losing the paper and never saving the file end the same way.