The main difference between a Gay's sexuality for men and a man's sexuality for men is Gender. Let me explain this for a layman.
A man (or a straight man, if you please), when he desires a man (even if straight men don't do it openly because of the societal pressures), like him as a man -- that is they think of them as a man liking another man. This doesn't make them feel any different from other men, and he doesn't want another identity from other straight men for that. This is beside the point that all straight men have a strong sexual need for other men.
When a Gay likes another man, he does it so thinking he is a woman liking a man. In western societies he is made to think that his liking for men flows from his femininity, but that is basically wrong, because more than 90% of feminine or queer males are into women rather than men.
The Gay feels he is different from other men whom he calls 'straight'. Now, in a society which confuses gender with sexuality, he thinks it is because of his sexuality for him (which is termed as 'homosexuality', when actually its 'heterosexuality' because he is seeking a masculine man, when he himself is feminine).
Also, just like in India, feminine males don't think of themselves as 'women' even though they feel like women, they have a different identity -- third gender, so they think of themselves as a distinct gender from men and women. Likewise, gays too feel like they're women, but consider themselves a distinct identity from both men and women, which they identify as 'gay' or 'homosexual' (i.e. they wrongly explain this difference in terms of 'sexual orientation').
This is the reason, straight men don't identify with the concept of 'homosexuality' or with 'homosexuals' or with gays -- even when they themselves don't know the reason, they instinctively rather fight with their sexual feelings for men, rather than be considered 'homosexual'.
The concept of 'homosexuality' has integral links with male femininity, so homosexuality, although defined as a 'man's sexual feelings for men', actually means a 'queer's sexual feeling for men'. And so it can't be used to describe a man's sexual feelings for men (or a straight's sexual feeling for men), even if technically it fits the description. Just like, if you start calling 'sun' as the 'moon', then the actual 'moon' cannot be called 'moon' anymore, because then it becomes problematic. And in the case of gays, it is thousands times more problematic because of the stigma attached to male femininity (queerness).
Also, although the Western society says gender is not important, its only outer sex that is important, the fact is that most gays look for 'straight' men or at least 'straight-acting' men for sex -- (which means masculine acting, not really someone who pretends to be into women). And they do complain that there are hardly any gays who are good enough 'straight-acting'. Because, acting cannot be as good as the real thing.