I believe that a key element in this debate is the fact that bisexual people do not belong to the two groups that society views as "integral" to the definition of sexual orientation: in short, they are somewhere in the middle between homosexual and heterosexual, without ever truly "belonging" to either community. Thus, since they cannot be easily defined by traits or qualities linked to a specific group, they are regarded as anomalous in their behaviour, incapable of integration and potentially deceitful.
While I imagine that this can be very painful, it is also undeniable that there is an element of truth in every stereotype - in fact, stereotypes are born from communal behaviours that are prevalent enough to be exaggerated and systematically used to define entire communities, both negatively and positively. However, in the case of bisexual people these stereotypes are magnified because of the perceived "otherness" of their sexual orientation, especially in the case of men. Bisexual women are treated less harshly because, in our patriarchal society, their sexuality is viewed as just an accessory to male sexuality. No matter what they do, perceive or feel, women are considered to be there to serve men. Only women who defy notions of proper femininity (they cultivate a manly appearance, seek power or non-sexually conditioned attention) are punished by society. Women who comply with the "desirable-submissive-happily inferior" concept of femininity, are viewed as lacking power and thus, they don't pose a threat to anyone.
Men, on the other hand, have it harder. Not being fully heterosexual is shameful because men who are attracted to other males, betray their self-appointed superiority by being "feminized", which is shameful because women are considered to be inferior. On the opposite side, the gay community has had to fight very hard to achieve visibility and lend validity to the idea that homosexuality is not a learned behaviour that one deliberately chooses to engage in, or the result of corruption.
Bisexual people, who can selectively enter both worlds, are seen as giving arguments to those who think that same-sex attraction is just an abnormal whim, or a sign of indecency. Half-homosexual, the heterosexuals view them as potentially deviant sexual predators who are incapable of honouring a "formal" relationship and complying with acceptable models of emotional and sexual interaction. Half-heterosexual, the homosexuals view them as exploitative sexual predators who can enjoy the best aspects of being gay, while rapidly retreating to comfort of straight relationships when things become intense, complicated, or potentially conflictive. Thus, they are bound to be rejected by everyone.
This is undeniably sad and unfair, but it's also true that these views, no matter how negative, are based on reality - obviously, not everyone's reality, but a reality frequent enough to create very negative perceptions nonetheless. Challenging these perceptions can only be achieved through increased visibility, showing everyone that bisexuality is not subjected to generalized moral imperatives, but individual behaviours. There are despicable people everywhere, even though societal pressures do play a very important role in the relationships that bisexual people have with the other two groups. Could it be possible to say that bisexual people are, often, cornered into hiding their true desires out of fear of rejection and isolation? I think it is. Society is very cruel with those who are different, and prejudice is a driving factor for many interactions. It's also true that, being in the middle, they may never completely rid themselves of the 'they can go the other way at any moment' cliche.
With this said, I think that women would be more likely to date a bisexual man, because women are conditioned to be more forgiving and accepting - even when being forgiving and accepting is painful and damaging to them. Also, as someone already mentioned, women are taught to hope they can change the men they enter in relationships with. Men are not taught to tolerate, in most cases. Women are taught to sacrifice themselves for the sake of a relationship, because they are derided and pitied if they don't have a man in their lives. "Alone" is a synonym of "failure" for women.
Gay men might be willing to sporadically have intercourse with these men, but the idea of making extreme concessions isn't something that most would accept (nor should anyone, no matter what sexual orientation might be). No one likes to feel like they are being used to pass the time until that socially acceptable partner turns up. This tends to elicit great anger among gay people, rather than resentment or jealousy (?) over the fact that bisexual people can blend into heterosexuality without problems. Equally, no one likes to be told that their sexual orientation makes them less deserving of love or acceptance - there are bisexual men out there who genuinely believe that they are better than gay men because they can potentially enter heterosexual relationships, and shamelessly objectify homosexual people. While biphobia seems to be prevalent among many, which is unfortunate, many people also seem to be overlooking the fact that some bisexual people are rabidly homophobic themselves.
Educating people in theoretical principles is an important matter, but only exposure, respect and honesty can ultimately be used to bring acceptance for misunderstood communities, in my opinion.