Size
The two phones are nearly identical in size. The only significant difference is that the Droid Maxx is nine percent thicker than the svelte GS4.
Weight
There are heavier phones than the Droid Maxx, but a featherweight it is not. It tips the scales at a full 28 percent heavier than the Galaxy S4.
Build
The Droid Maxx sticks with the kevlar build that Motorola has been using for its Droid Razr line for the last couple of years. Here it's covering more of the phone than ever. (kevlar is a really strong and tough material)
Display
Both displays give you the same spacious 5-inch layout, but the Galaxy S4 is much sharper. The Droid Maxx's 294 pixels per inch should make for a pretty sharp screen, but that's still a fairly low number for a high-end 2013 Smartphone.
Processor
The Motorola X8 is the branding for the system-on-a-chip in the new Droid phones, but the processor is (according to Ars Technica) a slight variation of a Qualcomm Snapdragon S4 Pro, which was featured in several high-end handsets in late 2012.
Who Won
So the GS4 wins this round. But let's keep this in perspective: both phones blow away the fastest phones from just a couple of years ago, and will be plenty fast for most users.