Is Judaism more friendly to the LGBT community than Christianity?
Not necessarily. It depends on which Judaism and which Christianity. Some Christians are fully accepting of gays as are some Jews. Some Jewish congregations are strictly opposed to homosexuality, as are many Christian ones.
The Bible is quite unequivocal in its condemnation of sex between men, transvestitism and the confusion of gender. Orthodox Judaism therefore, takes a rather firm stance on homosexuality, bisexuality, transgenderism and transvestitism (although, there is little concern about lesbianism, this would not fit easily into orthodox Judaism either).
Reform Judaism accepts, even celebrates, homosexuals but it can only do this by essentially ignoring the Torah. Most Reform Jews argue that the Torah is just a guideline rather than the final word on how Jews should act and do not consider it "the revealed word of God".
The choices for Christians are similar. If Christians consider the Bible the "revealed word of God" they are more likely to condemn homosexuality. If they think that Jesus effectively nullified the Bible (am I right in believing that's a thing?) then there's no problem with gays.
A lot of Jews, even many orthodox Jews, feel compassion for gay people. The science is really telling us they were born that way. There is a struggle around these issues even within the orthodox community. The most liberal orthodoxy can get however is "don't ask, don't tell" but the frum lifestyle is not good for gays. A good Jew is expected to get married and have a minimum of two children and gay sex is totally forbidden, so gay orthodox Jews are likely to suffer.
Personally, I'm inclined to think that denying gay people their right to find partners is unsustainable and unjust, but I also acknowledge the dilemma for the religion. Even though I do not see the Torah as a book that was literally revealed, I am also disinclined to treat it with disrespect. Tradition matters, but so do people's feelings and their happiness. My personal choice would be to welcome gays, accept them as they are, not preach to them about their sexuality, but also not celebrate gay weddings in synagogues or make a big deal out of homosexuality either way. The people matter, what they do behind closed doors is between them and God. Why make life harder for these people? We should make life easier. I, personally, don't think the Creator of the universe is really so petty as to get hung up on how people express their love for each other, but, then again, there could be a lot I'm not seeing. The best is to leave it up to the conscience of the people involved and love and support them no matter what- that's what makes sense to me.
Outside of religion, in secular societies, gays should have full legal rights, including marriage rights, but, in religious matters, our rabbis also have the right not to participate in such marriages. Secularism therefore gives us a way to have our religious cake and eat it too. We can vigorously defend the rights of the LGBTQ community in the broader, secular society, while maintaining the integrity of our religious laws within our own congregations.
I don't know how many Christian groups also see things this way but it seems like a reasonable way to proceed. Thank god for secularism!