It's kind of a big question.
Playing card designers will try to seek whatever edge they can in the marketplace, especially as there are more and more Kickstarter deck projects. One of the ways a designer will do this is to create a Bicycle-branded version of their deck. There's a subset of playing card collectors who specialize in the Bicycle brand, sometimes to the exclusion of all others. A subset of this subset will also strive to collect EVERY Bicycle deck that gets made, whether they're USPC-designed and -produced or produced by a third party company by USPC. Not every deck gets to be Bicycle-branded - it does have to be approved by the brand manager, and certain things are off the table, like decks with Satanic imagery, nudity, etc. But those that do can often see a slight jump in sales as a result of having that brand name.
What determines the size of a print run? There's a number of factors. One of the biggest factors would be the minimum size of the print run permitted by the printer. USPC won't make decks with runs of shorter than 2,500, Expert PCC's limit is 1,000, Legends has a limit of 900 and MakePlayingCards.com will make a run as short as a single deck - but the per-deck cost goes up the shorter the print run is. The designer's goal is to make a deck with high retail value and high aftermarket value - basically, to make it desirable. A deck's rarity can be a factor in determining its popularity with some collectors.
Some designers will intentionally make a deck in short supply, producing a pre-fixed number set before launch. Others will let the launch itself determine the number of decks to be made, making enough to meet the demand of the project's backers and perhaps a small amount extra for personal use, retail, wholesale, etc. Serially-numbered decks have a certain appeal to collectors as well - it's a feature that costs extra, and low-numbered decks might fetch a certain premium. In cases where a designer is creating a variety of deck designs, some will even produce numbered sets where all the decks in the set have a matching serial number in their individual series.
Now, as far as the value of a Bicycle-branded deck versus the unbranded version of the same deck - that's entirely up to the market! Sometimes one version or the other will be made in shorter supply, so that's probably the biggest reason why one version would sell for more, especially in cases where the only difference between one deck and the other is the Bicycle branding. In cases where both decks are in equal quantities, sometimes the Bicycle version will go for a little more, but more often than not, the price will be comparable if not identical.
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