Hopefully, I'll be able to answer this as I'm not so familiar with the Other Guys.
The Other Guys, that is, Montessori Schools, have always been Waldorf's "arch enemy." The feud goes down in the books along with the Leafs and the Sens. The Jays and the Tigers. Trump and Rosie. Reagan and Saddam.
But really, it's just a different way of looking at education.
Before I get into that, however, let's do a quick crash course on each school.
The Waldorf School was founded in 1917 by German philosopher Rudolf Steiner in Stuttgart, Germany. Waldorf was originally intended for the kids of the factory workers at the Waldorf Astoria Cigarette Company. Meanwhile, Maria Montessori opened the first Casa dei Bambini (house of children) in Rome, Italy in 1907. Montessori was a medical doctor and anthropologist.
Now, back to the hard stuff.
The philosophies. Steiner used anthroposophy as the groundwork for Waldorf's teachings. In simplistic terms, anothroposophy is the belief system that before someone has an understanding of the universe they must first have an understanding of humanity. This teaching is set throughout the Waldorf educational system and all the schools currently continue to follow its rules.
On the other hand, Montessori has no real set spiritual backing even though Maria did have some type of idea or philosophy on how her school should be run. However, unlike Rudolf, Maria never trademarked her thoughts as a set rule to all of her schools, therefore, some Montessori schools differ in how they educate their students.
A quick story which will obviously lead to another difference between the two educational giants. When I started at the Waldorf school coming from the public school system, I was shocked at the school's view of popular culture. Some students never owned a TV. Some never wore name brand clothing. Some even didn't have computers. Waldorf schools and preschools believe in progressing the child's mind through nature and art. Creating rather than copying. On the flip side, Montessori schools - even though they do believe in educating the whole individual by motivating his or her mind, body and soul - lets the child's parents make more decisions about what's right for their child.
Ultimately, when it comes to the actual learning process, the two schools are very similar. Both schools work in multi-year cycles to develop the child's mind. Waldorf works in seven year cycles, while Montessori does six year cycles. Both schools follow hands-on and intellectual ways of learning. Both schools have a "hippy" outlook in life while building the whole child, teaching them to think for themselves and how to avoid violence.
These teachings, even though very idealistic and capable of bringing much peace to the world if more people were to follow it, puts a mask on the children who are lifers at the school. They're not aware what "real life" is about. Of course, it's up the parents to teach their children balance between the ideal world and the realities of life.
I hope this quick little comparison taught you something about both schools to give you a better idea on which school to choose. However, it's all up to you. As Waldorf and Montessori teaching would say, you have to judge for yourself. Both schools have ongoing open houses. They both have reading material on the net that you might want to check out. Most importantly, though, it all has to do with your child.
Some kids are better fit for one school and not the other. Some kids, like me, just learn to appreciate the school they're in. Ask your child. Let them experience both and let them make up their own minds. That is after all, both the Montessori and Waldorf ways.
Quick Fact: Both schools were shut down during WWII because they refused to teach the ideology of the State.