I've been playing it since final version launch and have been really enjoying it.
Gameplay
If you really enjoyed Frontier: Elite 2 then you will really enjoy this new one, however if you didn't enjoy the older game you will not like this either. Also do not expect anything new. As it currently stands the game is near identical to Elite 2 in terms of what you can do and the actions you can take, except you can't currently visit planets as you could before. For the moment this are just giant planets you can orbit and look at, but not fly own into and explore the surface of. This will come later.
Controls
The game is really hard to control from a keyboard and mouse and is not recommended at all. Using an Xbox 360 controller with some keyboard control does work well so it's a fairly cheap control solution. However, to get the most from the game you need a dedicated HOTAS flight controller. I'm using a Saitek X52 Pro with the game and it is perfect. There is however a worldwide shortage of many of these controllers at the moment due to this game, so you will have to hunt to find one. They are also not cheap.
Docking
Gone (for the moment) are docking computers. It all had to be done manually and is as tedious as ever. Lets hope docking computers make a return. But once you get the hang of docking it isn't so hard. There is however a kind of bug a lot of us have had trouble with, and this is orientation of the ship when finally landing on the docking pads. If your ship isn't facing the correct way on the pad you bounce of and don't dock. Took everyone some time and frustration to work that one out.
Navigation
The intergalactic star map for navigating is very similar to before. However you no longer have direct information on star systems as before until you actually visit them, or purchase the map data from a station. So you are flying blind until you expand your travels.
As the game world is persistent (see below in trading) you no longer have access to speeding up time, as you did in Elite 2. You therefore have to fly in realtime and do everything as it happens. Thankfully you can enter fast travel to move across systems quickly, and jump between systems, so it isn't as bad as it first sounds.
Trading
This at the moment is less enjoyable than it used to be. Before you could look at the different station price lists and work out a good supply run to start making some profits. Now however this is not possible because prices fluctuate all the time depending on supply and demand. You might find something selling for a good price in one system, but return again later to find a surplus of that item due to a lot of traders selling there, so it no longer forms a good profit run. This is down to the persistent world model now used in Elite. Even playing single player the game needs to be connected online because it updates the game based on the multiplayer universe, as as prices fluctuate online they also change for you locally. Same for the game universe over all. So nothing is ever as predictable or the same.
And due to all this it takes a lot longer to make any money, and therefore upgrade or change your ship.
Graphics
The biggest difference over previous game is the graphics. Just as Elite 2 really moved the game on from its wireframe origins and no longer relied on your imagination to see the 3D objects, now the game really fleshes out the universe. The ships look really nice, the space stations look very realistic, and space just looks amazing.
Finally Dying
If your ship was destroyed in Elite you always had the option to just load up a saved game. However, now because the game using a persistent game world (even in single player) you don't have game saves. You live in the world and can't go back (unless you start a completely new game). Instead when your ship is destroyed you have to pay for a replacement to continue. If you are short of funds this means a loan and you start owing money in game. You can purchase insurance to pay for this when it happens, and to protect your upgrades.
I think that's it for a short summary.
I would highly recommend playing around with the offline tutorials to begin with the get used to the control system, navigation, docking etc. It makes a huge difference. The play single player for a bit to get used to putting it all together and getting a hang of flying and navigating. If you jump straight into the multiplayer game expect to not last long if you venture outside of the safer inner systems as pirates will ambush you and other players will target you as you approach stations to dock.