Solid - maintenance-free versus inflatable (air-filled) wheels - what wheels to choose when buying a stroller?
TL;DR: Air-filled wheels are still the most comfortable option for uneven surfaces or all-terrain use (think of bicycles). They, however, do have their drawbacks, compared to no-maintenance foam-, rubber- or plastic-covered wheels, mainly their weight, the need for inflating, and the risk of a puncture. Solid wheels are lighter and often more agile but wear down quicker and often can't cope with rough terrain conditions.
When choosing a stroller, stroller system, pram, pushchair, or buggy, you need to evaluate many features and aspects, all with respect to your lifestyle and needs. One of the more important points that come to mind very quickly is wheels. Stroller wheels are a very visible part of the chassis - and often crucial for the daddies, so make sure you know what he thinks before choosing ;-) Wheels come in all sizes (not shapes, though), and there are several types to consider.
The first polarization in the wheels topic you will notice is between the wheels with air-filled tires and everything else. Firstly, they are often making the stroller heavier. Then there's the thing about having to exercise certain maintenance with inflatable wheels - the inflating from time to time, for one, and changing the inner tubes in case of a puncture. Not even talking about the risk of damaging the wheel when inflating it... if you overdo it, you can bend or crack the wheel rim, and the whole wheel is unusable (needs replacing). That is why I'll strongly advise you against inflating your stroller's wheels on a gas station, on those strong air pump machines meant for larger wheels of a bicycle - a breakage often happens right there.
So why are air-filled wheels still made and wanted if there are such negatives to take into account, you say? Well, it is because their driving characteristics are still, very often, the best. Why do you think bicycles are made pretty much ONLY with air-filled tires? Inflatable wheels are maneuverable, much more terrain-capable, they add a certain amount of natural suspension to the ride, and they are larger than most. That's always good on holes, bumps, stones, even if only around town - you won't feel a bad surface as much with large, soft-riding wheels.
Other types of wheels are called no-maintenance wheels. Don't be fooled, please; every wheel needs a bit of maintenance here and there (checkup, cleaning, lubrication)... But foam- or gel-filled wheels (usually with tyres filled with material, not air), rubber-covered or rubberized wheels, EVA (Ethylene Vinyl Acetate) foam wheels, and plastic-covered wheels don't need inflating (and there's no risk of a puncture).
The most all-terrain type would be any wheel with a tire, no matter what it's filled with. There are certain risks of a very harsh ride; the inside could "leak" out of you tear the rubber of the tire... but this pretty much never happens, so no worries. Such wheels are, however, on the larger side, and the rubber with something inside is the closest thing to air-filled wheels, maneuverability-wise. The foam inside is, in my opinion, and experience, the better option - the best brands in the world go for a foam-filled tyre, usually... that is primarily because the gel is heavier (and a little bit leak-prone, compared to foam you find in, e.g., Joie Versatrax).
The rest of the wheel types is quite straightforward:
Rubber wheels are solid wheels of a higher quality than EVA foam-covered wheels, but they may cause a more 'rigid' feel of the ride. It's good if rubber-covered wheels are accompanied by a soft-ish suspension. Then they can be very capable even in harsher conditions.
EVA foam wheels are solid wheels covered with foam and not the best for rough terrain; on the other hand, they are very agile, maneuverable, one of the reasons for this being their low weight. They are the most lightweight of all, which will best suit urban kind of use. Foam wheels are also usually the most budget-friendly.
The only other basic type encompasses cover all-plastic (hard of softer feel) wheels typical for old-school buggies. You can find modern, soft plastic wheels with some brands still (like those on Quinny Zapp X line). They are comparable to EVA foam wheels, just not catching stones inside the outer layer... but mostly, 'plastic wheels' is a negative connotation for something cheap and rattly.