Heir to the British throne Prince Charles has married divorcee Camilla Parker Bowles without a constitutional crisis. When Edward VIII wanted to marry Wallis Simpson he had to abdicate.
Of course Charles himself is also divorced.
Was this purely a result of differing social attitudes at the time, or were there other factors that made Simpson unacceptable to the government of the day?
It is mostly due to the differing social attitudes of the day, but the legal position was also different in 1936. The Wikipedia page is pretty clear about the social attitudes, but I'll try to explain the legal issues here.
In 1936 the Church of England opposed remarriage after divorce. Furthermore, at that time, the Church of England considered adultery to be the only grounds for divorce. In English law, grounds for divorce were governed by the Matrimonial Causes Act (1857). In brief, men could sue for divorce on the grounds of adultery by their spouses, while women had to prove additional offences (such as incest, sodomy, cruelty etc.) had been committed.
Wallis Simpson's first divorce (granted in the United States) had been on the grounds of "emotional incompatibility", and so was not recognised by the Church of England. In the eyes of the church, therefore, her second marriage was bigamous. There was also the very real possibility that had her divorce been challenged in an English court, it might not have been recognised under English law.
In 1937, a year after the Abdication Crisis, Parliament passed the Matrimonial Causes Act (1937), which expanded the grounds for divorce, but which would still probably not allowed Wallis Simpson's first divorce to be recognised had it been challenged in an English Court.
The law required that the monarch must be in communion with the Church of England (the monarch was head or "Supreme Governor" of the Church of England). A bigamous marriage would create a split with the Church as precipitate a constitutional crisis.
Since 2002, the Church of England has allowed divorcees to marry. Furthermore, both Prince Charles and Camilla Parker Bowles had divorced under English Law (which had become much more liberal since the Matrimonial Causes Act (1973)), so there could be no legal challenge to their union.
Charles and Camilla were therefore able to marry without any problem in 2005.
One thing I'm not seeing in the answers so far is that having been married prior wasn't the only strike against Wallis Simpson. She:
All of this just did not appear to add up to a person who was liable to take her responsibilities as Queen Consort of Britain as seriously as the position requires. She could possibly have even been a security risk, at a time when war looked likely.
I think the Wikipedia Article 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 have changed a lot in the 70 years between Edward's abdication and Charles' marriage to Camilla. Today divorce and remarriage are widely accepted.
In 2002 the Church of England officially allowed divorcees to remarry. (Article in the Telegraph). This paved the way for Charles and Camillas marriage without any problems in 2005.
A rather more surprising argument is advanced here: https://www.lrb.co.uk/v10/n16/paul-foot/the-great-times-they-could-have-had
Wallis was, as was popular at the time, a Fascist. As the country drew closer to conflict this became increasingly intolerable to the security services:
In all the innumerable versions of the ‘Greatest Love Story of the Century’ it is assumed that the British Establishment, led by Stanley Baldwin and the Archbishop of Canterbury, could not stomach the idea of a monarch marrying a twice-divorced woman. The objections, it is said, were moral and religious. The truth is, however, that throughout the centuries archbishops and prime ministers have miraculously overcome their moral objections to royal idiosyncrasies in the bedchamber. The real objection to the liaison between the King and Mrs Simpson was that both were Nazi sympathisers at a time when the more far-sighted civil servants, politicians and businessmen were beginning, sometimes reluctantly, to realise that British interests and German interests were on a collision course. As the biographers of Baldwin, Keith Middlemas and John Barnes, observed, ‘the government had awakened to a danger that had nothing to do with any question of marriage.’
Charles Higham quotes an FBI file in Washington: ‘Certain would-be state secrets were passed on to Edward, and when it was found that Ribbentrop’ – the German Ambassador in London – ‘actually received the same information, immediately Baldwin was forced to accept that the leakage had been located.’ Higham then asserts (without quoting the relevant passage): ‘The same report categorically states that Wallis was responsible for this breach of security.’ Of Sir Robert Vansittart, Permanent Under-Secretary at the Foreign Office and head of British Intelligence, Higham writes (and here he does provide the evidence): he ‘was Wallis’s implacable enemy from the day he was convinced she was a Nazi collaborator’.
