I think the [Wikipedia Article][1] on this is quite clear and comprehensive.

The key points:

- Edward was king and as king, he was head of the church, which at the time didn't allow people to remarry as long as their former spouse is still alive. Having the head of the church violate this rule is clearly unacceptable.

- Prince Charles is not the king, therefore he is not head of church and wouldn't be bound by such constraints, though likely he would have had to renounce his claim to the throne, if he had done this in Edwards time.

- social conventions changed a lot in the 70 years between Edwards abdication and Charles marriage to Camilla. Today divorce and remarrying is widely accepted.

- In 2002 the Church of England officially allowed divorcees to remarry. ([Article in the Telegraph][2]). This paved the way for Charles and Camillas marriage without any problems in 2005.




  [1]: https://en.wikipedia.org/wiki/Edward_VIII_abdication_crisis
  [2]: http://www.telegraph.co.uk/news/uknews/1413238/Divorcees-can-now-have-full-church-wedding.html