Price, weight and features are the biggest differences in the two.
Pros for Infinium: its lighter - I must use a Minelab ProSwing 45 when I use the ATX on a sandy beach or land. In the water, no problem. The Infinium can be hip mounted, above, below the shaft or under the forearm. Less than half the price of the ATX. Less maintenance, especially the shaft. The ATX's shaft and cams require extensive cleaning after a salt water hunt, fresh water - not so much on land, it is almost even with the Infinium cleaning requirements. Interchangeable coils at 1/3 the cost. Listed waterproof to a depth of 200 feet. - ATX is only to 10 feet. Less water resistance while swinging due to the physical size of the ATX's larger coil, shaft and control box, especially if you hip mount the Infinium's control box. Easier battery access (1 screw on cap verses 2 twist lock) and only 4 verses 8 AA batteries required in the ATX.
Pros for the ATX: more control of the detector's operation including a volume control. This makes up for many of the Infinium's Pros. I also believe it signals on smaller and deeper gold rings. BUT, as I detect the New England area beaches, fresh and salt water, my impressions may not hold true for other beaches as the likes of those in Florida, Caribbean, or Pacific.
I use Gray Ghost Amphibian headphone of each detector and never have increased the volume to full on. Batteries get recharged after 11 hours of detecting - with either detector. Both detectors remember the last settings when turned off, unless the controls are changed. But MUST be set up for different areas (beach make-up)
I have had my Infinium and all of the available coils for 8+ years, my ATX with only the standard DD coil for almost 2 years. Now my ATX is my primary go to salt water detector. The Infinium is mainly used on winter beaches and shallow water in among the rocks with an 8" mono coil or the small 5" elliptical in river beds. For fresh drops in the summer time...I use an Exclaibur or ATPro.
I hope this helps and keep in mind: it really comes down to the fact that each detector/coil combination has its strengths and weaknesses for a given environment. The more you know of the conditions you are detecting and the strengths of a detector needed for the objects your are searching for...there is a "BEST" combination out there somewhere. The ultimate choice is yours. Make it and learn the detector well and you will be rewarded.