I'm trying to merge 2 datasets (both shapefiles, but I can convert them to raster layers if necessary) where the values are compared, and whichever is higher becomes the new value for the layer. How would I got about this? They aren't quite adjacent to one another, because there's a fairly large overlap in the middle. I'm using ArcMap 10.1
To do this you would first Union (keep all fields) the two shapefiles together and add a numeric field called say MaxValue.
Then, assuming you have values from shapefile1 and shapefile2 called val1 and val2 respectively you can use the Calculate Field tool with the settings below to calculate MaxValue: