Conventional medicine is science and evidence based. Before a drug, surgical/medical procedure, or diagnostic test is incorporated into practice, it is studied and determined if it is safe and effective or will find the problem for the diagnostic tests. If a drug, procedure or test fail to show efficacy and safety, it is discarded and will not be used in practice.
Conventional medicine is also constantly evolving. New and improved medicines are always being introduced, making treatments more safe and effective. New diagnostic and imaging tests are being developed making it easier to detect diseases. New surgeries are also being done. Prosthetic limbs and organ transplants are examples of this. A practicing doctor will need to always update himself or else he will risk being left behind and outdated.
Homeopathy on the other hand was developed in 1796, a time when medicine was still mostly mumbo jumbo. It is based on the principle that like cures like, but only if it is diluted so much that not one molecule of the substance remains. This goes against everything we know about chemistry, physics and pharmacology.
Homeopaths do not have a good theory on how their remedies are supposed to work. They will often invoke quantum physics and water memory. Water memory has never been proven and quantum physics is just used as a cop out reason as even physicists have a hard time with it. In short, homeopaths have no idea how their remedies are supposed to work.
Homeopathy has also been studied for effectiveness but has failed to show any positive results. Homeopaths will claim some studies have shown positive results but those studies have been found to be of poor quality with flaws in methodology and with bias.
edit: There you go, one of the answers here described "energy level" and "body level" while at the same time neglecting to define what those are. That is another difference between medicine and homeopathy. Medicine has precise and measurable definitions. Homeopathy is stuck in the mysticism and unproven and undefined concepts such as water memory, potentiation, succusion, energy level, and body levels.