Badiさん、こんにちは。 あいだに is simply used for setting a timeframe. うちに has the same meaning + few more, and one of them is the usage for this case; to make the causal relationship between A and B. e.g. 電話をしている　うちに　　子供がいなくなった 　　　電話をしている　あいだに　子供がいなくなった Whereas あいだに simply mentioned "while I was on the phone",　the sentense of うちに here means "Because of A:電話をしている, B:子供がいなくなった happened", in English "I lost my kid because of the lack of attention while I was on the phone". Your sentense of 日本にいる　うちに　一度富士山に登りたいです seems to contain the expression like "As far as I am here in Japan, I'd like to climb Mt.Fuji because it's symbolic and well-known, and once I left, I don't think I'm back only for climbing", so we could say that うちに in your sentense gave more motivation of you to climb than the another. Your sentense of あいだに is also correct, but if it's compared with うちに, あいだに is just used for being predicated such as 'during my stay in Japan'. I hope this helps.　いつか富士山に登れるといいですね！

その2つは同じ意味です。 「うち」も「あいだ」も色んな意味があります。 この場合の「〜うちに」と「〜あいだに」は「during」とか「while」という意味です。 There is almost no difference . There is various meaning "うちに" and "あいだに". That it's this case, "〜うちに", “〜あいだに”, the meaning as "during" and "while".

For the first time, you can consider that they have no difference. نفسه