I have lived in Beijing for 2 years now and I have seen many people come, study the language for a few months and then leave. When these friends come back for a quick visit, naturally, they are worse at Chinese than when they left. The pattern that has clearly emerged is however this: people that have practiced speaking, not only studied the language, are far better off.
There is no clear neurology behind this. The truth is that scientists have not nearly sufficient knowledge of the brain to give a deterministic answer to why and how we are able to learn stuff the way that we do. What can be discerned through empirical studies is however abundantly clear; when you know how to ride a bike it stays with you. Even people with dementia can peal an apple. The actions that we learn stay with us far longer than the data, facts or vocabulary. Theories we hammer in while burning the midnight oil, hovering over a textbook is not as firmly logged in our brain. The main reason, in my mind, that people do so atrociously in secondary school language studies is that there is no exposure to real culture there. The theory we learn there is just that: theory. It cant compare to actions.
When people come to China this is even clearer: some people take the initiative to engage the country, some dont. The ones that do progress quicker, and the knowledge, which is actually more skill that knowledge, stay with them longer. The body is a fabulous tool, and the brain is wired into it, of course, and when actually using your mouth and ears, your actions are wired into your memory banks the same way. For this reason, learning Chinese should be more like learning to ride a bike than it is today.
Most schools just use the hammer. The reason for this is clear. It is really really expensive to teach in another way. Chinese is different from Spanish, French or English. The tones baffle most new students and it takes a lot to over power this barrier to learning. Many hours of classes focused only on pronunciation will take its share of any budget for the simple reason that one teacher cannot teach thirty students how to pronounce the tricky tones.
It needs to be a one on one, or at least one on few, classroom environment for students to be able to wrap the tongue around a Chinese pronunciation.
This is the main reason why so many university students in China come out of a year of study with very little real life skills. If they would have studies with a proper academy, not all are proper (!), they would have had the face time with a teacher, early on, that is necessary to learn proper pinyin.
I studied with a great school before I got my present job (as a recruiter for language schools) and the first thing they taught me was pronunciation. It has helped me immensely in my studies because when I knew how to speak (real) Chinese I was not developing a list of vocabulary, I was developing a new skill set, in a very real way. Sure, it took a lot of time, 3 hours a day for 2 weeks, with just me and 1 teacher. But what I got out of that cant be measured. It transformed learning Mandarin as a theoretical subject and made it into pealing an apple or riding a bike; my learning process has been defined by action and for this reason, as with the friends that have studied in similar ways, my Chinese is action based and will stay with me longer, hopefully for life.
Tones matter because they have the power to transform the learning process. Dont use the hammer, the midnight oil and a large class size. Use your mouth and your ears. Learn tones!