They are not always synonyms. The usage of надо/нужно: 1. I need to do = lit. To me needed = Мне (dative case) надо/нужно + inf. Мне надо работать сегодня вечером. I need/have to work today in the evening. 2. I need smth = lit. To me it’s needed = Мне (dat) нужно/нужен/нужна/нужны + noun (nominative case) нужно/нужен/нужна are short adjectives, so you have to agree them with the noun. Present Мне нужен зонт. Ей нужна сумка. Past Мне нужен был зонт. Ей нужна была сумка. Future Мне нужен будет зонт. Ей нужна будет сумка.

They are absolutely interchangable. Надо is a little bit informal, and being so, used a bit more frequently, but such a little bit that it doesnt make any difference. You can use that which is easier to pronounce :D

but if you use it in an official letter it is better to write - необходимо

You can use both. Those are synonyms =)