ArcGIS treats two similar projections definition differently

I am about to recreate a global grid in ArcMAP, based on the Behrmann projection I though.
The map I want to recreate has this definition (in the .prj file):

PROJCS["WGS_1984_Cylindrical_Equal_Area",
GEOGCS["GCS_WGS_1984",
DATUM["D_WGS_1984",
SPHEROID["WGS_1984",6378137.0,298.257223563]],
PRIMEM["Greenwich",0.0],
UNIT["Degree",0.0174532925199433]],
PROJECTION["Cylindrical_Equal_Area"],
PARAMETER["false_easting",0.0],
PARAMETER["false_northing",0.0],
PARAMETER["central_meridian",0.0],
PARAMETER["standard_parallel_1",30.0],
UNIT["Meter",1.0]]

Which I though was the exact same as ArcMAPs Behrmann projection:

PROJCS["World_Behrmann",
GEOGCS["GCS_WGS_1984",
DATUM["D_WGS_1984",
SPHEROID["WGS_1984",6378137.0,298.257223563]],
PRIMEM["Greenwich",0.0],
UNIT["Degree",0.0174532925199433]],
PROJECTION["Behrmann"],
PARAMETER["False_Easting",0.0],
PARAMETER["False_Northing",0.0],
PARAMETER["Central_Meridian",0.0],
UNIT["Meter",1.0]]

But ArcMap treats them slightly differently and can project one to the other. And they don’t exactly line up. What is the difference?