Traditional medicine (predominantly herbal medicine) is based on a philosophy of health care. It is based on the premise that given the right environment, bodies will heal themselves. It also seeks to find the balance that is right for that body, and is mostly about getting people back to optimal health.
Modern medicine (allopathic medicine) is more a system of disease treatment. It is about correcting imbalances by forcing the body back to the norm, eg surgery removes diseased or broken parts, drugs can correct a hormone imbalance by supplying the appropriate hormones (traditional medicine works on the appropriate organs so the body rights its own hormone balance), or causes changes that will correct a blood pressure problem.
In traditional medicine, diagnoses are made based on what the body is trying to do, eg a fever is telling us that there are pathogens present, and the heat of the fever is to resolve the infection. By suppressing the fever we are stopping the body from doing its job. If the body is not strong enough, the fever will be ineffectual, and traditional medicine practitioners will help the body to have a good strong, but short fever so that the infection is resolved. In allopathic medicine, antibiotics are given to kill the pathogen, and do the body's job for it.
Traditional and allopathic medicine, although having very different philosophies, work well together. For instance, someone who is taking antibiotics can also be treated by a traditional practitioner so that the antibiotics have less adverse effects, and the body is strengthened so that it is not so susceptible to infections in the future. (There are times when antibiotics are absolutely essential for saving lives).
People undergoing cancer treatment can have allopathic treatment like surgery, chemotherapy and radiotherapy, and at the same time the traditional pratitioner can treat them to reduce the side effects of the cancer treatment, or prepare them for surgery.
Allopathic medicine works best when there is clear pathology, whereas traditional medicine works well even when people are just "out of sorts" and there is no clear pathology.
Advantages of traditional medicine are that it is cheap, and is empowering to the patient. It recognises that not everyone fits the "norm" and allows for individual differences. It helps people regain health, and is not restricted to treating diseases.
Advantages of allopathic medicine are that it has tools to keep people alive so that they may then be able to regain their health. It is brilliant for emergency situations, accidents, and "putting people back together". It is also great for diagnosing disease states. (In fact, in some parts of the world, traditional practitioners are forbidden by law to make medical diagnoses)
This is a very brief answer to what is a potentially huge essay question. In a nutshell, traditional medicine is good for getting people back to health, and allopathic medicine is good for keeping people alive. If I get hit by a bus, I want to be taken to an allopathic hospital, but then I will see a traditional practitioner afterwards to make sure I am not dopey enough to get hit by a bus again.
I hope this gives some insight into the differences between traditional and allopathic medicine, and wish you luck on your quest.