I first would suggest trying the 14.20 BETA driver first before you start distribution and release hopping, if 13.352.1014 doesn't work for you.
You might also be able to get the open source driver to work if you have libmesa compiled with elf-utils and llvm+clang for the gallium3d API support.
You might also want to see exactly which kernel version you have in Fedora 20 (if it's not 3.14) and edit the patch as mentioned...:
Code:
--- lib/modules/fglrx/build_mod/firegl_public.c
+++ lib/modules/fglrx/build_mod/firegl_public.c
@@ -1784,7 +1784,11 @@ KCL_TYPE_Uid ATI_API_CALL KCL_GetEffecti
#else
#ifdef current_euid
+#if LINUX_VERSION_CODE >= KERNEL_VERSION(3,14,0)
+ return __kuid_val(current_euid());
+#else
return current_euid();
+#endif
#else
return current->euid;
#endif
...and edit the highlighted part to reflect the EXACT version your kernel is.
Example if you use Kernel 3.14.4, you use code
KERNEL_VERSION(3,14,4)
Good luck.