Depended a little bit on the exact vehicle. Spare track links could be found on the bow of a Pz III or Tiger, or the sponson rear on a Panther. Spare roadwheels for the PZ III or Pz IV similarly had mounting points (and unlike a US tank, the mounting points for these roadwheels were specifically designed. Indeed, you can tell the approximate date of manufacture of a JgdPz IV by the location and type of the spare roadwheels even if you can see little else of the vehicle). However, given the sheer diameter of Panther or Tiger roadwheels, it became a little difficult to find real estate on the tank, let alone physically lift the thing up into position. (Today’s roadwheels on an M1, for example, are aluminum, and still not exactly light). Plus, of course, the German tanks come well equipped with anything from pioneer tools to lifting jacks, all of which have to have space allocated for them on the sponsons, so it’s not really fair to say that they had no equipment. Take a look at my video of the Panther exterior, https://www.youtube.com/watch?v=wKKktWu6Rh0 or the Panzer III, https://www.youtube.com/watch?v=Jdlcj6JTdzw, you’ll see they don’t actually have much room to carry much more.

There are a couple of extra issues with the US though, which make them seem more festooned. One is, quite simply, that US tankers, not having to carry anything themselves, and at the end of one of the most amazing supply chains in existence, could both acquire additional equipment, and carry them around. The other was that Allied tankers often felt under-armored, and were more than willing to add track, logs, sandbags, roadwheels, or anything else they could mount between themselves and the opposition. German crews started doing the same thing, for example with concrete or logs, but there were two more differences. The first was that, for whatever other sins the German heavies had, Panther and King Tiger had sufficient frontal armor that there was little need to add to it, and the other was that they listened to their engineers, and the Americans didn’t. The boffins on both side of the Atlantic (what is American for “boffin?” “Egghead”?) concluded that these field expedient armor loadings actually caused more harm than good, but with the notable exception of Third Armor, no US commanders enforced the recommendation. The thinking may well have been that if the troops felt better they would fight better, regardless of what the engineers said. The Germans, however, seem to have enforced it, and field added armor became very rare. As a result, Allied tanks in WW2 often looked much more like Fury than not. (The real Fury actually didn’t have it, and was a 75mm tank, but, hey, movies...)

As a side note, US tankers still to his day festoon their vehicles with kit. If you compare an M1 with any other tank, Challenger, Leopard, T-72, you will see large bustle racks for personal gear, often overflowing. The kit bins on other tanks tend to be much smaller, if they exist at all. A US tank unit in the field bears more than a passing resemblance to a gypsy caravan...