I'm sure that all the SH regulars will know this backwards and forwards, so this is more for the benefit of newer people, or people who stumble in via google, or people who want a quick link they can throw out as an answer to anyone who asks the question.
So, what's with the goofy-ass road wheel design on German WWII AFVs?
A puzzled and terrified worker struggles to comprehend and assemble the suspension of a tiger I
You may have run into a variety of explanations for this running gear design; that it provided a smoother ride, that the design saved rubber, or possibly some other rubbish. Like the myth that frontal drive sprockets provide more traction (seriously, how in the hell is that supposed to make any sense?), these wrong explanations of the merits of interleaved road wheels seem to rise from some quote taken out of context.
The interleaved road wheel running gear may have saved some rubber relative to an alternative design that was particularly wasteful of it. But interleaved road wheels are not particularly economic in this respect because, and I realize this is a complicated concept to explain so I'll try my best, they have more wheels. Interleaved road wheels do allow for large wheel diameters, and a larger diameter wheel will spread wear out over a larger circumference. So interleaved road wheels might allow for the rubber on the wheels to last longer, although their construction would require more in the first place.
Interleaved road wheels would not improve ride quality either. The ride quality of a tank is not a function of the size or number of wheels it possesses, but of how they are sprung. So, it is possible that in certain competitive trials an interleaved road wheel design outperformed a design that lacked this feature. I could readily believe, for instance, that the tiger (H) had a better ride quality on rough terrain than the tiger (P), or that the SDKFZ. 251 had a smoother ride than the M3. However, this would be because the tiger (H) and SDKFZ. 251 have independently sprung road wheels on torsion bars while the tiger (P) and M3 do not.
Torsion bar layout of the tiger II
Volute spring suspension of the M3 half track
So, what do interleaved road wheels do?
They have two principal effects; one is a small benefit, and the other is an enormous detriment.
The small benefit of interleaved road wheels is that they spread the weight of the vehicle out more evenly on the track links:
The weight of a tank is not completely evenly spread out on the contact area of its tracks. This is because tracks are not rigid. If they were, they would be mainly ornamental and tanks' engines would just be for show. More of the weight of a tank is concentrated under the parts of the track that the road wheels are sitting directly on top of. Additionally, once a tank starts to sink into the soil a bit, larger road wheels work better than smaller ones because the larger ones have more contact area. But you can only fit so many large diameter road wheels in the space of a tank's hull.
So, the only way to have lots of road wheels and have big road wheels at the same time is to interleave them. Simple as that.
If you would like an exhaustive look at the development of the semi-empirical MMP equation, read this.
The major, crippling downside to interleaved road wheels is that it makes changing the road wheels extremely time consuming.
A pair of workers perform maintenance on a panther tank, and contemplate the futility of all human achievement
Lucas Friedli reprints in his book on big cat maintenance a report from a training unit complaining that replacing the inner road wheels of a tiger tank took ten hours. That is completely outrageous, and was a contributor to the poor operational availability of the big cats.
For this reason, interleaved road wheels have rarely been used after World War Two; only on a few French prototypes and a Swedish APC:
PBV 302 variant with interleaved road wheels
Some bizarre French tank