There is no direct opposite to the Suzuki method, but teachers commonly refer to Suzuki vs. "traditional" teaching. Since the latter term is vague, for the purposes of this discussion I will coin the term "Smedberg method" after my own violin teacher, whose methods are decidedly far from Suzuki. She breaks down every aspect of violin technique and musicality in terms of cause and effect, with the goal that her students learn precisely how to create the sounds they desire. Because her method of teaching is so analytical, she rarely accepts students under the age of nine, as they can hardly be expected to understand these concepts presented in such detail. It's the analogue of studying language by learning grammar rules and vocabulary lists, which one would never try with a young child. ("What's a past participle, Mommy?")
The "Smedberg method" has been wildly successful for me, no surprise because I am a highly structured thinker with much training in analytical fields such as physics and computer science. For the first time in my life, I feel like I'm beginning to understand the relationships between my hand and arm motions and the sounds that come out of my instrument. It used to be the case that if I couldn't get the result I wanted even after a lot of practice, I'd simply say to myself, "Well, that's yet one more violin technique I'll never get." Now, I've learned to narrow down the problem. Is it my left hand or right hand? Is it pitch or tone that's off? Can I do it correctly if I play it very slowly? No longer is successful violin playing a result of some magic that one has or doesn't have; it's all a matter of techniques that can be analyzed and mastered.
These are the same principles that serve me in my language studies. I learn the rules of verb conjugation, noun declension, adjective agreement, mix in a healthy dose of vocabulary, and I'm on my way to constructing whatever I need. I know when I'm supposed to use the Spanish subjunctive and which German prepositions take the dative case. With a dictionary, I can theoretically use these languages perfectly. And yet, there's a catch...
The danger in relying too much on the Smedberg method of violin instruction, or the classroom method of language learning, lies in that little word: "theoretical". Just about any structured discipline can be analyzed to death, but ultimately these concepts must be put into practice. Although my teacher and I can---and often do!---talk endlessly about planes of motion, centers of gravity, difference tones, and the harmonic series, none of it means anything if I can't translate that knowledge into beautiful violin playing. Because theory appeals to me so much (need I specify that my doctorate was in theoretical physics, and that I'm a danger in the lab?) it's tempting to spend my entire lesson talking about the violin instead of playing it. Similarly, one cannot ever hope to achieve fluency in a language if one is constantly going through the mental process of verb conjugation and vocabulary retrieval; at some point, it has to become automatic.
So the key to language success, I believe, is finding a balance between theory and practice. Too much practice without any theory---generally the plight of people who learn a language through pure immersion---leads to error-riddled fluency and dependence on what "sounds right" without knowledge of why it's right. Too much theory without enough practice---the problem faced by those who study language assiduously in school but never use it outside the classroom---leads to error-free but painfully slow conversation. Start with the aspect that appeals to you most, or is most accessible to you, but do not neglect the other.