The philosophies differ the most in early childhood. Waldorf will have your child engaged in imaginative play. Montessori will focus on preparing your child for academic learning. Here's an oversimplification, but it's a start: Do you want your child to play with dolls at ages 3, 4 and 5 or with mathematical cubes?
Waldorf elementary will focus on hand skills, like knitting. Montessori will include 'practical life' skills, such as cooking, but will focus on academic exercises that are presented on trays or in little kits. The kids are supposed to engage the materials, discover what the teacher wants them to discover, then put the materials back.
Both W and M use a set curriculum and have a relatively rigid idea of what 'success' looks like. In Waldorf, success is a relaxed and imaginative child, a little free-spirited nymph, free of commercial influence, who follows the teacher willingly. Believe it or not, that puts a lot of pressure on a family. It's hard to raise a commercial-free child, and not every child is totally submissive in the presence of a teacher. In Montessori, success is an academically organized child who can plan work and see it through to completion. To me, that looked great on older children, but a little sad on the little ones. I was not convinced that academic work is the best use of early childhood.
Neither is particularly successful at integrating children who don't fit the model. At East Bay Waldorf, many boys are shamed for their natural boy energy. For example, kicking a ball is a no-no at the Waldorf school. Every Waldorf school is different, and some are more welcoming of boys than E.Bay is. At Berkeley Montessori, kids from more challenging backgrounds have not had their needs met, even though that's what Montessori was designed to do. The children they serve best are the children of professors and other white-collar professionals. Although I have to say that Berkeley Montessori has a very professional team that can address learning challenges and emotional issues.
You can see that these philosophies start out on very different paths, yet to me they are very similar. They are both have a lot going for them, and they are both rigid. They are both interested in certain results that preclude other possibilities.
There are other choices out there that don't ask you to choose between imagination and academics. Gay Austin has a wonderful play-based preschool that offers all of the imagination of a Waldorf school along with just the right amount of kindergarten preparation. In elementary schools, there are a lot of inspiring choices that are not rigid philosophically. Aurora, Escuela Bilingue, Prospect Sierra, Windrush, to name a few. Walden is an creative alternative to the Waldorf school. Amongst public schools, NOCCS and Canyon are interesting. And there's a public Waldorf-inspired school in Sebastopol. There, they would be forced to balance play and academics because of the state standards.