What core mechanic do cards provide in tabletop card games and what is the equivalent in a computer game?

In a tabletop game, cards provide random characters and situations (and sometimes, outcomes). In a computer game, you can simply introduce a character, who introduces a situation and has AI.

I think cards can be thought of as a "super die".

A stack of cards offers a controllable, tuneable randomness, not limited to 6 sides (or any number of sides). You can have as many cards in the deck as you deem appropriate, and as many duplicates as you wish, not being limited by a physical property (other than what fits on the table). You can even add "expansion packs" later if necessary (which means extra $$$). Further, a card can hold much more information than just a number, which is greater convenience for the player (no 10 minutes browsing tables in the DM's book).

Lastly, a stack of cards introduces strategy and player skill. The experienced player knows that there are only 2 moles and 2 vampire rabbits in the deck. Vampire rabbits give a +30% win chance on everything but moles, whereas moles instantly kill rabbits but lose to everything else. So, after you've played two moles already, the experienced player knows that it's safe to play a vampire rabbit.

The idea of a physical paper card as a collection of all the stats related to one combatant predates CCGs and TCGs by decades. Baseball cards have been around since 1860, and data for weaving patterns and library books was also stored in systems of cards since approximately 1890; card catalog, rolodex or filing cabinet systems (when there is too much data for a single card) have also been used to organize data about people in military, medical, law, and business applications. A card is a piece of paper, and the concept of a magic spell (for example, a spell to summon a monster) being stored as writing on paper or a tablet (like a proto-card) is about as old as writing itself. Even in our mostly-paperless era virtual note cards are a popular brainstorming and idea-organization method because the concept "one rectangle of words and pictures = 1 idea" is just so natural to humans. So, that's my cultural answer to "why cards?"

For a more functionalist answer, the whole mechanism of giving the player a random resource at the beginning of each turn is contrived, artificial difficulty to make games more surprising and replayable. If the player is assumed to be a mage of some kind, it would be more logical to assume they would have access to their entire library of spells at all times and be able to choose what they want. The physical properties of a deck of cards make this randomness seem reasonable, whereas most other types of storage would not. If the spellcasting process is imprecise a mage might try for a type A demon and get a type B demon, but it makes no sense that a mage who wants to cast one kind of spell could accidentally cast something completely different. If you are storing summon spells in your mind, in a book, in an array of balls/capsules/boxes, whatever, you would store them in such a way that you could quickly grab what you wanted, or at least something similar.

Quarriors is one example of a card system having been partially replaced with a dice system, but it still uses reference cards because the dice are too small to contain all the data or nice graphics. In a computer game a die could be large, but then would any one face of the die really be different from a card?

I've played with board game developers and had a blast using placeholder objects according to a set of rules in place for the game. Could have been done on a screen or whatever, or some bottle caps, toothpicks, a piece of paper, and dice. There were definitely mechanics based on the rules.

From the strictly physical side, as already noted by others, nothing beats cards for presenting detailed information about an individual object; in fact most displays of information about one of many similar objects turn into pseudo-cards in videogame user interfaces, even if they aren't used as such. Therefore using cards or not depends only on having objects that are suitable for a card metaphor because they behave like cards: accessing sequentially the elements of random permutations of a large set, using them as a discrete and coarse-grained resource that can be accumulated (usually to offer the players the option of doing more or better moves when they matter in exchange for doing little in other turns), exchanging them for other cards or comparable objects, hiding and showing them, etc.

I want the mechanic but do not want to be bound by the aestethic, so what's the mechanic?
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But on a computer that's just an aesthetic choice, not a mechanic.

I think you are muddling things up a bit here.

Cards are a form of presentation that suits some designs and game mechanics better than others. Even in physical games there are no mechanics that absolutely require cards. For example, picking a random subset of a collection of objects is actually more convenient with a set of solid chips and an opaque container than by shuffling a deck of cards.

On a computer, card presentation is not just a matter of subjective taste. It's a matter of usability, accessibility and affordance. When some objects in the game behave like physical cards, showing them as such helps the player in understanding the game space. Conversely, you should probably avoid such presentation for game objects which behave in ways fundamentally unlike physical cards, even if you like the aesthetic.

The fundamental card property which the presentation conveys is immutability. The card itself does not change over time and has no internal state. This is tremendously helpful for reasoning about the system as a whole (see: functional programming).

The secondary card property is that a card can only exist in one discrete state at a time, visually indicated by which zone it occupies in the game (such as "in hand", "in deck", "attached to card X on the table", "on table and face down", "discard pile"). Card mechanics either move cards between these states, or apply passive effects from cards to elsewhere in the game.

As mechanics go cards are inferior in computer world (you can do everything card can by other means better on a computer), except for the mood. There seems to be a relatively strong attachment to cards (as representation of things on the screen) I was inquiring players and made some topic here and generally the "cards idea" had a warm welcome.

I would say the best things of cards is to actually show cards on the screen. People seem to like it.

Case1: A card game, or a board game with cards. Result: it is just easier to use cards and nobody thinks twice.

Case2:
We look for iconic figures, for stability.
Gambling and money are tokens that frequent card use. Although they are not the staple of stability we're quite looking for, they do mean something, fun. These cards are more fun for people who use them frequently.
We can quickly imagine the complex rules a single card governs, and its importance, just like in reality our gambling and credit cards have so many rules. So slap on whatever rule you want, someone who reads it is less likely to laugh at it and throw it away.

We trust our icons to a fault.

Promise of collection raises obsession in goal oriented individuals.
Having many different cards creates a bountiful story, even if there's a single passage on each card we hold. People love stories, and can't always accept an ending, maybe this can emulate a good book.

You simply want to reach an established fan base. Maybe even you yourself have a bit of a card addiction and thought this was the greatest idea in the world. (Ok, maybe it is, just look at past examples, it's a working business model).

Result: Suddenly you've made a pinball game with cards that mechanically shuffle for visual and sound effects.

As mechanics go cards are inferior in computer world (you can do everything card can by other means better on a computer)

In a computer, rand is basically our dice.
Let's say that in a game, you've got a 50% chance to hit, so on a regular dice you'd have to roll a 4/5/6 to hit or 1/2/3 to miss (or in a computer, use rand).
The alternative with cards is you make a bunch of 'hit' and 'miss' cards, and put the same number of each into a deck. When you draw a card you put it into a 'used' pile, which is reshuffled once the whole deck has been used.

There's a *huge* difference between these two approaches, which a professional game designer will be aware of: the amount of statistical variance in the outcomes.
Flipping a coin and getting 10 heads in a row is extremely unlikely, but if you flip a coin all day long, it will eventually happen. Likewise, if enough people play your game then it's a statistical fact that one of them will roll misses for their first hundred rolls, will be killed by the first beetle that they fight a few times and decide that your game sucks ;)
Decks allow you to put limits on the variance, e.g. At the extreme, your deck only has 1 hit and 1 miss card, which guarantees that you can't have a miss streak longer than two misses.

Slot machine designers use both the raw probabilities and the variances ruthlessly to tweak different games to different demographics, for great profit.