The Reich lacked the capacity to produce enough trucks to handle the support of motorized columns and the manpower to run them. Without sufficient high speed transport the armored columns were always right on the edge of fatal supply shortages. This was especially the case in the East as the Heer advanced farther away from its home base.
Heer policy was to build supply depots every 100 km or so, usually starting near a rail line. A truck on paved road, or dry ground, can do a round trip a day over that distance. Horse drawn wagons will take at least six days for the same round trip (and that pace will kill the horses in fairly short order). Couple that with the fact that a truck can, in most cases, transport more material per trip than a wagon and the problems multiply.
Fodder for the horses is also bulky and uses more space in rail cars than fuel, although fuel is heavier (weight is less of an issue, especially with rail cars than cube space). It was impossible for the Heer horse herds to "live off the land" since each division deployed between 2,500 and 6,500 horses at any single time (the Wehrmacht, including both the Heer and Luftwaffe ground element, ran through an astounding 2,750,000 horses during the war, most of these had to be transported by rail to the combat zones). A German infantry division would have around 4,000 men dedicated to the task of teamster or otherwise supporting the livestock according to its TOE.
The best "tale of the tape" is military truck production during the war (Note that WAllies figures include vehicles provided to the USSR under various "Lend Lease" efforts by both the Commonwealth and U.S.):
Canada - 815,000
Britain - ~740,000 (figures vary)
USSR - 195,000
USA - 2,382,000
Total - 4,080,000
Nazi Germany - 346,000
To put the Reich's figures into perspective, the U.S. provided ~240,000 medium/heavy trucks as well as ~160,000 other motor vehicles, or ~ 400,000 vehicles just to the USSR. I wasn't able to find a figure for Commonwealth contributions.
In summary, the Heer was either pressing the offensive (which always uses far more supply than defending) or was retreating against an enemy with air superiority (in the East) or Air Supremacy (in the West and in the East starting in mid 1944), which results in massive losses in logistical resources, from June 1941 to the end of the war. In a way its a miracle that the entire Nazi war effort didn't simply flip over onto its back like a bug and die by late 1943.