(Mostly Monsey here, as it’s where I live 🙂 )
Schooling is hard in both places. The more on the modern side you get the easier it is, but even so, even the more moderate yeshivish places, like YSV, are very crowded (there is a freeze on the number of classes in the boys’ school, and it’s mostly siblings and children of alumni). The chassidish places are apparently impossible.
Home prices depend on the neighborhood. I know little about Lakewood, but in Monsey, housing has been getting more and more expensive, especially in the main areas. Home prices on my block, for example, have quite literally tripled over the last fifteen years. In some areas they’ve quadrupled. If being near shopping and many minyanim is important to you, be prepared to shell out. There are outlying neighborhoods which are cheaper, though, so if that’s an option for you, look into it. House sizes depend a bit on the neighborhoods, but there can also be a variety of sizes in the same neighborhood, depending on when it was built- some blocks are developments, some are older…
Commuting times to where? Monsey is doable to Manhattan, Lakewood is much farther.
I believe that you need cars for both. Monsey has very limited public transportation largely dependent on neighborhood- there are several areas where public transportation is a reasonably viable option, but it’s really not common.
Monsey’s also very safe. (Ramapo was the second safest town in the country in 2011!) There has recently been some crime, but it’s still bH very rare.
We’ve got public parks, decent (if crowded) infrastructure, mandated school busing, the Palisades Mall, some nice new restaurants… it’s a cool place 🙂