I have played the original Elite on the C64. Also V2 on the Amiga, followed by Frontier. I loved them, so I am a bit biased.
But I've also played quite a few other games involving both space combat and trade, like Privateer, Freelancer, and EVE Online, so I think I have some perspective, too.
First thing I loved in this game, is the space combat and flight model. Even simple things like docking the space station can be fun. Or being able to slow down using gravity or nearby plants. Power distributor - you can go for ENG for increased speed and maneuverability, WEP for maximum firepower, or SYS when you want to recover shields quickly - dynamically in the combat. You can enter silent mode and disappear from enemy scanner. You can also use Flight Assist Off mode, which allows you to shoot and fly in two different directions :). Very nice pilot experience overall.
Graphics and sounds are great. Stars, planets, ships - polished, beautiful. Sometimes I just stop to enjoy the visuals. Game is nicely optimized, runs smoothly on my GTX 660.
Bounty hunting, piracy. You can go either way - be a bad guy and take cargo from traders, or hunt those with bounties on their head, solo or with help of local security forces. Or you can be a pirate in federation, and good guy in the empire space. Or enter the anarchy systems, without any laws. Up to you.
Missions are simple. No campaign, or anything like that. Deliver a message, transport a cargo, smuggle something in, find some specific things in space, kill pirates. I usually do them as a side tasks - when trading on high-profit route I check board for trading missions, when bounty hunting I check missions requiring visiting unidentified signal sources. Sometimes you can meet a ship who gives you counter-proposal. When going for combat missions be very careful with high-bounty combat missions (those 180k+), as killing Elite Python or Anaconda might be pretty challenging task, especially when it gets plasma accelerators.
Multi-player in current version is pretty basic. You can chat, you can interact with each other, but nothing like corporations or sovereignty known from EVE. Full freedom, for better or worse, depends on what you like. Make friends or enemies, help or shoot anyone you want - your choice :). I could use a group chat for easier communication with friends, though - I hope to see this in 1.1 or 1.2 patch (support for Wings is coming).
For anyone who like the idea of space combat/trade/exploration sandbox without too many rules, this will be a great game. It is simple, but it's also polished, and can be really fun to play :).