Can someone tell me why Bruce left in the first place?

In the official Iron Maiden biography, Bruce explains that the seeds of discontent were sown as far back as 1986, when the band were getting together ideas for 'Somewhere In Time'. Bruce apparently felt that Maiden had taken their sound/style at the time as far as as it could go and needed to do something a little different, or risk stagnating and becoming a parody of themselves. He wanted to do a semi-acoustic album and came up with some 'folksy' sounding songs. However, Steve and the others rejected these songs out of hand. Steve was quoted as saying he thought Bruce had "Lost the plot". If you look at the album notes for SIT you will not see a single Bruce Dickinson writing credit. Bruce said that the feeling of rejection left him 'Squashed on the inside for a long time'.

In the years that followed, many observers commented that live, Bruce seemed to be slowing down and not giving 100% with Maiden any more. Then when his solo album 'Tattooed Millionaire' and the subsequent tour became a success, it seemed to prove that he could survive without Maiden. Ultimately, it was his solo career, radio work and other interests outside Maiden that made him decide to leave. He simply wanted to do other things.

Adrian left because he was unhappy with the direction that Steve Harris was planning to take with 'No Prayer for the Dying'. The band wanted to make a really raw sounding album like 'Killers'. Adrian felt that Maiden were heading in the right direction with SIT and Seventh Son... and that it would be a mistake to strip their sound down again. I also understand that the band had decided to start recording before Adrian had a chance to contribute any ideas.

danimal97520 that's *beep* Susperia85 got it 100% correct and was confirmed by Adrian more than one time after he reunited with Maiden. Plus, there's one song in the NPFTD album credited to Adrian.