To be honest, I have not used hands-on myself any of these two software packages since 3 years ago, but traditionally Fluent (now belonging to ANSYS) has been the leader CFD in aeronautics applications and for computing the aerodynamic performances of racing cars and wind turbine rotors because of its superior performance (accuracy of the solution, computational time needed, etc.)
CFX was the CFD of ANSYS. Then ANSYS bought Fluent, and this probably is enough justification as per which of the two software packages is more successful. I do not know the particular reason why ANSYS still offers the possibility of using CFX. Maybe its a strategic reason or CFX is better at solving some particular fluid problems. You may want to check this in view of what you want to solve. So, if you can choose, I would recommend you to use Fluent.
What I don't recommend you at all is to use Gambit (the CAD\mesher with which Fluent used to work; actually, I think it has even been removed from the market). There are far better CAD and mesher software packages available.
I used two different CFD software a couple of years ago. I could be wrong but I think the CFX is more reliable on a combustion problem and the Fluent is better for hydrodynamic problems. Also, the geometry drawing scheme is different. I think CFX is more graphic user-interface structure I guess. However, I don't know what is the better on two different cases. Because, the companies merged a couple of years ago. I believe there some communication to improve each software. This is all I know.
I have used CFX for the past 5 years and I think it is an excellent product. I have also used Fluent but do not have the same level of experience or knowledge. However, CFX is an evolving software that seems to improve with every edition, which makes it more user-friendly, and this doesn't seem to happen at the same rate with Fluent.
One of the main aspects that you should consider is your intended application. I have used CFX for hydrodynamic problems (e.g. wave-structure interaction) and heat transfer problems and it handles these very well.
Additionally, if this is the first time you have used this type of software, it may be slow to learn. You should investigate what suitable tutorials (ANSYS Help, YouTube etc.) are available for your application and for which software, CFX or Fluent. I started with CFX when there wasn't many tutorials or support and it was a slow, yet worthwhile, process when learning it initially. This has nothing got to do with the performance of the project but may be valuable if you are undertaking a masters or have limited time while completing a PhD study.
CFX is much reliable in case of results but in Fluent you can change input for some specific application oriented problems...I general CFX is better with good post processing phase