It depends somewhat on your card. Mesa (open source drivers) doesn't support all of them and not every game run on mesa (Alien Isolation, Dying Light and Divinity Original Sin) but generally the user experience is a lot better with mesa. Framerates might be a bit lower.
In short: Install drivers from their website and think if open source drivers would be enough for you. Later change to open source if you feel that you can live with it.

I'd recommend to use the open source drivers if possible, as they are a lot less buggy than fglrx and work well together with the screen configuration tools of desktop environments, something that cannot be said of fglrx. (With fglrx don't even think of using your desktop's screen configuration tools, as already the thought of using them might break your fglrx installation permanently, causing graphics artifacts to appear in games and during video playback. Only use Catalyst Control Center or aticonfig.)

But, as tumocs said, the very latest cards will not work well unless you are running Linux 4.5, and even with the very latest mesa, kernel and llvm versions some games won't run, as the open source drivers currently only support OpenGL 4.1.

If you want to use fglrx, I'd strongly recommend to stick to the packages supplied by your distribution, as they are tested and are already packaged for you, so you can easily uninstall or upgrade them whenever needed.
If you cannot do this for some reason, do under no circumstances install the driver version from the AMD website directly, as this can and will cause issues when you either want to uninstall it, or ever want to upgrade your distribution to a later release. Always tell the driver installer to create distribution specific packages (the --buildpkg command line option), and then install those packages using your distribution's package manager. This will allow you to uninstall or update the drivers in a clean way.

I am new to Linux so i have a question as well. i had linux mint 17.2 installed and upgraded internally to 17.3. But since then the fglrx driver in the "driver manager" didnt change version wise. Is that so because AMD abandoned the fglrx for their new AMDGPU driver scheme and because of this no new driver come out?
Or is it so because i didnt make a clean install to Linux Mint 17.3?
Or does the "Driver Manager" even download new drivers or is it just there for choosing different prepackaged drivers with the distribution?
Or are the Linux Mint providers not updating their "own" fglrx driver (on their repository) and they are the cause?

I asked in the Linux Mint Forum but unfortunately nobody answered at all.

On a sidenote what is the difference between fglrx and fglrx-updates because they are the same version and all.

Distributions usually ship older "stable" versions of the proprietary drivers. Linux Mint would likely ship drivers from 2014 (since Mint is based on Ubuntu 14.04 AFAIK). If you want the newest FGLRX you need to download it from AMD's website directly but keep in mind this can compromise your distribution's stability.

Another thing is on amd.com if i search for the driver there is a linux and a ubuntu driver which one should i choose? Because Linux Mint is based on ubuntu i think the ubuntu one but i am not sure.

On a sidenote i installed the ubuntu driver and now my linux mint is broken :(

its ridiculous that installing a simple uptodate gpu driver can break the Linux so easily and they expect noobs to handle console commands in this day and age :( i am installing SteamOS now instead of linux mint. At least i hope they can put a working self updating gpu driver on it. ARGH :(( i mean come on installing an uptodate gpu driver worked like a charm even on windows 95 (1995) loool

wanted to install steamos but it isnt working because for whatever ridiculous reason the debian install underneath needs a cdrom drive even when you install from usb flash drive. And i dont have any cdrom drive in my pc lol, so it seems i cant install it as well. loooooooooool

for anyone who has the same problem with steamos i found a working solution in the steam forums which i post here as well:

I've restarted my install so I could document this more accurately. Select No to "Load CD-ROM drivers..." and No to "Manually select a CD-ROM..." You will get an error. Select the option to execute a shell. Do it.

Type the following command. Basically you are querying what disk partitions are visible to the kernel.

# cat /proc/partitions

Your output will look something like this:

major minor #blocks name

8 16 976762584 sdb
8 17 976760832 sdb1
8 32 1953791 sdc
8 33 1952767 sdc1

Find the numbered partition that corresponds with the installation media. Use the #blocks as a guide; it should roughly correspond to the size of your USB drive. Mine for example is clearly sdc1, a 2GB USB stick. The other device, sdb is a 1TB hard drive where I will be installing Steam. As a side note for those unfamiliar with Linux naming conventions, sdc and sdb are the device names. sdc1 and sdb1 are the names of the partitions themselves.

Anyway, now that you know the device you want to mount, simply run the following command. sdXX corresponds to the device you found above, yours may be different.

# mount -t vfat /dev/sdXX /cdrom

Now the installation media should be available for the installer to use. Type exit and click continue at the Detect and Mount CD-ROM step.

SteamOS installed but it seems i have a black screen after the install only. Probable Cause according to google i have a AMD GPU which causes this problem after first start of SteamOS ^_^