For companies like Lufthansa, SAS and others - premium economy is closer to what Business Class used to be 20 years ago. You get an amenity kit, better food selection with real cutlery, nicer pillow and duvet, and a much nicer seat - usually in a 2-2-2 configuration.
The differences I've mainly noticed in premium economy are: electrical outlets, nicer in-flight entertainment, more legroom/better reclining, nicer head cushion, closer to the front (if that's a benefit) and that's about it. Amusingly, I've found on Delta the "premium economy" section occasionally extends slightly outside, e.x. the electrical outlets and better in-flight entertainment (but not the legroom) aren't only for premium economy but also 2 or 3 rows of normal seats beyond. Which is a little strange, since I would think Delta would use the opportunity to present that as an upgrade (paid, or free as a goodwill gesture) but I've only ever seen them labeled and sold as normal seats.
Both carriers in the process of upgrading their seats in business class and adding a true premium economy product, not what delta calls comfort plus, which is really just more leg room at the front of the economy cabin. The true premium economy product is much closer to the premium economy that has been available on other carriers (emirates, cathay pacific, etc...) for a while and has better seats, food is served on plates, etc... It's not to far off from what one might get in a domestic first class.