Modified from an earlier answer
Without getting too much into the weeds of Second World War military history, the Allies had a much more limited objective in the Ardennes (containing and pushing back a German offensive) than the Germans in Barbarossa (defeat of the Red Army, conquest of the western USSR, etc.). In the case of 1941, the German military establishment took a cavalier approach to the capabilities of their Soviet enemy and their thinking mirrored that of Hitler- that one swift kick would lead to a collapse of the Soviet state. The campaign planners anticipated that the winter would mostly be a mopping up operation whose troops would require a minimum of supplies.
This massive underestimation of Soviet capabilities led to a disaster in late 1941 as the logisitical system of the Wehrmacht began falling apart at the seams. Not only had the Red Army continued to fight despite massive losses in the various cauldron battles, but Barbarossa's logistical network was incapable of adequately supplying the German armies in the field. Instead of mopping up, German troops were engaged in offensive operations in November 1941 that ate up equipment and ammunition at a prodigious rate. In such a overburdened logistical system, winter clothing often had a lower priority than fuel or ammunition.
In short, the "titans" of the German General Staff were in a dire situation of their own making. Their planning for Barbarossa was careless and they brushed aside inconvenient details that suggested defeat of the USSR would take more than a single campaign. Victory over France in 1940, which few officers in the Heer had actually expected, had engendered a fatal overconfidence which dovetailed with other problematic components of the German way of war.
But in one small way, the German planners of Barbarossa managed to wring a victory of sorts postwar by foisting all the blame for the campaign's missteps upon Hitler. The already common trope in the West of of "Russia's General Winter" (which itself is not true- Napoleon's Grande Armee was a defeated force by the winter of 1812) lent this distorted and selective memory of 1941 a degree of legitimacy.
Fighting in cold conditions is not the difficult part, its advancing. Defending in the winter is certainly easier for a variety of reasons. Supplying advancing armies is very difficult but even harder through snow. Logistics are often overlooked but are a crucial component for such operations. Difficulties arise when situations change, thats when you will see the logistic/strategic capabilities of an Army.
The US was able to withstand the German attack but had trouble advancing through Ardennes in January aswell. If you check for the progress not a lot happened in Dezember and January in terms of movement therefore supplying the troops was not comparable to Moscow 1941, this was also against an enemy who was retreating and redeploying its troops. During January the first German divisions started pulling out of the lines to redeploy to the Eastern Front. [1] The next big advance into Germany started in March.
Often forgotten is that Allied Armies pretty much never had to deal with partisans or "resistance" movements in generall. The Wehrmacht with less motor equipment had to supply their armies deeper in enemy territory with partisans attacking supply lines et cetera. The Wehrmacht was an aggressor and occupant, obviously the civilians were less willing to help unless forced. The Allied forces were liberators and could use civilians labor more easily if neccessary.
The notion that the Allies "had no supplies" is kinda a myth and get misrepresented. Allied unit had less supply than they had normally, that doesn't mean they didn't have supplies. A US division had far higher supply tonnage than a German division. When people talk about the supply situation in the Ardennes they often have the well known Siege of Bastogne in mind. The battle is well known due to its coverage in BoB and best seller books. Keep in mind the siege lasted a week.
To add on this assumption that the US forces faired well in winter conditions. The US forces lost more than 1000 [2] tanks within a month, highlighting that the US forces had indeed trouble to conduct orderly retreat in harsh conditions.
Since logistics and the general failure of the Wehrmacht in 1941 are a hotly debated topic its interesting to see that the Western Allies had trouble keeping the pace on after their breakout of Normandy. We are speaking of 300-400 Km before Pattons third army "had to" switch to a more defensive stance because the neigbhouring 1st army was given priority. [3] The US forces in the ETO were not believed, by their own command, to be able to conduct a full front offensive against ~20% of the 1944 Wehrmacht. The USSR was often advancing on the entire front over a far wider range. The US had to cut down supplies to one of their 4 armies because they wanted to focus their ressources on another one which ultimatly failed to fullfil the plan ( Advance around Hurten Forest). The US had by far the biggest GDP in the world but was seemingly unable to meet their own planned supply numbers. The US wasn't in the position to lose the war obviously, when their offensive stalled in October/November 1944 the Russian were already unstoppable. So their failure did not matter.
This was through friendly territory without any "unknown and unplanned" impact from sources like partisans or enemy airforces. The major difficulty was obviously that their supplies had to be brought via ship, but this was known beforehand. U-boats had minimal impact at this point in war and the Allied Armies had trouble fullfilling their numbers without any "suprises". Again this was against a beaten enemy who had the majority of its forces elsewhere. This should illustrate how difficult supplying huge armies really is. I think the Red Army should not be forgotten when it comes to the logistics topic. The Red Army was advancing very fast on full front against an enemy who was rarely willing to retreat. The Wehrmacht forces in southern France had a rather orderly retreat toward Germany and most of the Wehrmacht forces left after Normandy pulled back to the Siegfried line so most of the advances during late august and september were rather trouble free for the Western Allies. The Red Army on the other hand was considered the "main problem" and retreating without a fight was more rare on the Eastern Front. Still the Red Army to managed to advance big distance over long fronts.
[1][2] Christer Bergstrom: The Ardennes, 1944-1945: Hitler's Winter Offensive
[3] Zaloga, Lorraine Campaign.