Language Family Tree

Despite the presence of the word "linguistic" in the name of my college major, I still have no idea what makes one language family different from another. (A little research on Wikipedia reveals that my confusion is shared.) English is part of the Indo-European family. Khmer, according to my books, is part of the Mon-Khmer family. But in the end, they don't seem so different to me. I mean, yes: Khmer and English have many different sounds, and Khmer verbs are conjugated differently from English ones (that is to say, minimally). But in the end, the language is made up of words and sentences, and they're even generally in the order of subject-verb-object. What kind of acrobatic grammatical contortions did I expect I'd have to put myself through?

Maybe I am simply seeing the surface similarities because my current grasp of the language is so poor. Maybe once I am more conversant in Khmer, the language will fall clearly, in my mind, into a family of languages to which my native tongue does not belong. Until then, I will look at the language family tree with skepticism.