The word "queer" itself means across - it comes from the Indo-European root -twerkw, which also yields the German quer (transverse), Latin torquere (to twist), English athwart. ... The immemorial current that queer represents is antiseparatist as it is antiassimilationist. Keenly, it is relational, and strange.
- Eve Kosofsky Sedgwick, "Tendencies"
To me, queer is about celebrating difference and diversity while remaining related with other human beings. That's me.
The word "bisexual", on the other hand, has two etymological components, about each of which I have reservations.
Firstly, the word names sex rather than relationships - not surprisingly in a way, since it was popularised in a time when lots of theorising was going on about sexual categories. I suspect that part of the popularity these days of the abbreviation "bi" is that it sounds less like a medical/legal/social diagnosis and more like a culture. (According to that idea, Lesbian is to homosexual as gay is to homosexual as bi is to bisexual.)
Secondly, "bi", meaning two, is predicated on (a) the division of humankind into exactly two genders and on (b) that division being so massively important that special words are needed for it.
It's not only that there are various special words for the two halves of the binary (men/women, girls/boys, male/female, and hundreds more referring to either a male person or a female person in some particular role). There are also special words for people according to their sexual/emotional preference for one or other half of the binary (homosexual/heterosexual - though interestingly there are no common words for "person who prefers men" or "person who prefers women"). I.e. people decided that gender preference was such a central facet of sexuality that people had to be divided into categories based purely on what gender their partners were.
So the word "bisexual" comes with an odd history. It's the reuniting of two halves of a binary split, while carefully retaining the importance of the division. Two artificially separated halves of a whole get bolted back together again.
One understanding I've suggested of the diversity of bi people is a continuum with "both/and" bisexuals at one end and "either/or" bisexuals at the other. (See article on bi diversity.) While the colloquial names "both/and" and "either/or" are clearly still based on the mainstream male/female division, the "either/or" end of the spectrum could equally be named "transcending gender", since for people at that end of the spectrum, gender isn't an important criterion for picking partners.
So, it seems to me that for someone at the "both/and" end of the spectrum, since they do see gender as important, the word bisexual is reasonably appropriate. But for people at the "transcending gender" end of the spectrum, bisexual doesn't really fit as a name.
Since I personally am fascinated by the whole area of what becomes possible when gender is no longer important, I can't really align on the embedded meanings of the word "bisexual" as I can with the word "queer". In my intuitive sense of myself and my place in the world, "queer" is a much better fit than "bi".
However, even though it intuitively doesn't exactly fit, I still identify as bi, and my reasons for that are strategic.
Probably the most important one is the way that bisexuality is erased by monosexist assumptions. I know that if I were to identify only as queer, most people would assume I meant Lesbian. I want more people to recognise the choices available to them in this area. I don't want to be invisible. (And I also don't want any Lesbians to feel they've been led up the garden path - that way lies social disaster )
The other main one is that I'm interested in what people can accomplish in co-operation with one another, and there already is a strong and vibrant, albeit relatively small (compared to its potential), community under the name "bi". I don't want to reinvent the wheel by coming up with some brilliant new etymologically sound name and trying to recruit people to adopt it. I'm willing to build from what we've got.
In any case, I think it's a bit like band names - you can take any name ("Blancmange" is the one that always springs to mind when I think about this phenomenon) and after a while, if you hear enough of the music, that name begins to mean the music, instead of, or at least as well as, the original meaning. So even if "bi" originally meant something I can't align on, it now also means everything I know of "the bi community" - its people, its events, its literature, its ideas.