I'm thinking smooth finish -- the illustrations on my deck are slick and kinda non-frilly. I want the finish to be as clean as possible -- no dimpling. Seems like a pretty minority position these days, though. The only deck I know of that has a smooth finish is Aviators. Everyone seems to go with magic/air cushion these days. I guess that gives the cards a silkier feeling for handling/cardistry, but I can't help but feeling that the dimpling on the cards hurts the artwork.
Bee stock is thicker in most cases. It's the same paper as the Bicycle stock except for the thickness.
Don't confuse "finish" with "coating." A lot of confusion has developed over the years, and USPC does nothing to clear it up because they take advantage of the confusion when marketing decks. The finish of a paper is the texture - for playing cards, that's generally smooth or embossed. The coating is the laminate that goes over the cards after they're printed, which can make them glossy and helps increase slip and endurance.
Magic Finish technically isn't a finish, it's a coating. You can have a smooth-textured deck coated with Magic Finish. Magic Finish is the default coating for custom decks - you actually have to request the standard coating used on the basic cards you see on the shelf at the supermarket or the drugstore. Most people don't - while there's some holdouts that don't like the slickness of Magic Finish, it's a rare custom deck that isn't using it these days. The original codename for Magic Finish was Performance Coating, a name that Ellusionist continues to use today. After all, they made the first deck to use Magic Finish, the Gold Arcane deck.
There are a number of smooth decks on the market, some of which are even coated with Magic Finish. Besides Aviators, there's Streamlines, the Lo-Vision Bicycle decks, the Ivory Finish CARC Bee Erdnase decks, D&D's recent Steamboat reprints - and that's just off the top of my head, I'm sure there's more.
There is one noteworthy difference between a smooth and a textured deck. In my personal experience, if you take two decks, identical except for the texture - one smooth, one textured - you'll find that the smooth deck will be more firm and takes a bit more muscle to shuffle. My theory as to why has to do with the modern method of embossing a deck.
At one time, a deck's texture or lack thereof was applied in the card's coating (hence the confusion from the old marketing terms like Linoid, Cambric, Air Cushion, Ivory, etc.). This was done by using cloth rollers to apply the finish for a textured deck. But the cloth rollers wore out quickly enough that USPC, in the never ending quest for efficiency (meaning spending less cash), replaced cloth rollers with steel ones, creating the card's finish by pressing dimples into the paper for embossing and using smooth rollers for smooth decks.
This impression made into the paper of a textured deck weakens the structural integrity of the card, in much the same way that drilling holes in a wood board will make it weaker and easier to break. In terms of pasteboard and playing cards, that weakening allows the cards to flex more easily than a smooth deck - and conversely, smooth decks, with an "undamaged" surface, have greater structural integrity and are stiffer than their embossed counterparts.
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