Well, if I'm being cynical, I'd say that "premium denim" is any jean with a 350% plus mark-up. "Normal" mark-up for women's clothing is around 150% (which goes some way in explaining why stores can regularly offer 60-80% discounts without going out of business!)
Brand name, fit, style, fabric, finish, design, country of origin, and marketing all come into play. Most "premium" jeans stress the specialness of their fabric--which usually translates to how the brand and designer choose to finish the raw denim which comes from the mill. The more that is done to "finish" the fabric, the higher the price. Since many finishing techniques to soften and distress the fabric reduce fabric strength, "premium" denim can actually be more delicate than raw denim and might require more careful laundering. Special touches like decorative stitching, metal fasteners, and fashion-forward washes and styles can also command a higher price.
To my way of thinking, "premium" denim isn't necessarily better than regular jeans, any more than a $3000 leather bag is better than a $250 leather bag, or a cashmere sweater is better than one made of merino wool. All those bags and sweaters can be functional, "quality" purchases, and stylish additions to a wardrobe. Same for jeans. If you want a pair of quality jeans that fit well, look stylish, and last for several seasons of wear, you don't have to buy "premium" denim. If you want a special wash or finish, a fashion-forward style, and, a bit of the cachet that accompanies a brand with a good story or a coveted designer association--and are happy to pay more for your pleasure--then why not? Some of us indulge ourselves by buying a designer handbag or cashmere sweater, so why should it be any different for a pair of jeans?