As much as it hurts me to say it, the war in Burma was not important at all in the defeat of the Japanese. The war against the Japanese was decisively fought by the Americans. The Japanese were unable to supply and send reinforcements to Burma by 1944 due to the control that the Americans had over sea and air by then. On top of that, reinforcements were clearly more important to more vital locations closer to Japan. Burma did tie up a sizable amount of Japanese troops that could have been used elsewhere however, and Operation U-Go inflicted the largest defeat that the Japanese endured on land. In the end though, Burma meant very little in comparison to far greater and decisive battles in the Pacific, but it did mean a great deal to make up for the humiliation that the British suffered in 1941-42.

The second part of your question is also very interesting. Burma was not that important of a colony to begin with and was not prioritized for either supplies or reinforcements. Events in Europe overshadowed what was happening in Burma and it wasn't until 1943 in which good news began to come out of it. Events in Burma were always overshadowed by what was happening in Europe and while the British 14th Army was achieving amazing victories in Burma, the attention was instead aimed at France, Belgium, the Netherlands and Germany where the war was at its most "real" to the people on the British home front. Thus, the 14th Army became the "Forgotten Army".