Learn English – What does the phrase “50% premium” mean

Ordinarily a fine point means a subtle distinction (sense 8a in TheFreeDictionary).

The fine point isn't a standard term in any context, but per @Robusto's comment, there is the fine (or small) print meaning (usually unfavourable) details buried inconspicuously in small print at the end of a verbose contract. In common parlance it's often assumed that such apparently minor details may in fact be of crucial importance when deciding whether to sign a contract.

In fact OP has mistranscribed "The fine points" (at 1m 45s) in an instructional video. As @Spare Oom indicates, a fine point is a subtle or abstruse point. Used in the singular that would often imply an obscure, peripheral detail, but in this particular context the speaker is using the plural, and his meaning is...

The subtle details which must be mastered in order to swim really well.

This doesn't seem like a particularly common usage to me, possibly because of the unwanted negative associations with fine print, and a fine point in an argument, which is often said dismissively of some valid but peripheral assertion. In OP's context there are no such connotations or other ambiguities, but I wouldn't advise using the expression more generally unless your context is equally clear.

To augment Barrie's answer about the game, I'd say that a premium version of the game has, to use a cliché, all the bells and whistles than the stripped-down basic version. I don't play games, but my son does. Premium versions seem to come with more weapons, more options for each character and for each player, and other benefits that cost money. Premium (or paid) versions of antivirus and antimalware software provide realtime protection and automatic updates, but the basic free versions don't always provide that. A first-class airplane seat is a premium seat because it has more space, better food, better drinks, and better service than an economy class airplane seat. A premium version of an automobile has a bigger and better air-conditioner, leather upholstery instead of plastic or cloth or naugahyde (artificial leather), electric controls for the side-view mirrors, an electric radio antenna, maybe, a USB connector in the glove compartment for your MP3 player, a roof rack for your luggage and dog, a fancier radio & CD player, a built-in GPS, a camera for taking videos of your trip (maybe in front of the rear-view mirror), a little radar device in the front as well as the back of the car to let you know when you're too close to objects, maybe a couple of cameras to show you whats behind your rear wheels and your front wheels, etc.