What is the difference between “Gay” and “Homosexual"? Is it only by gender?

Both gay and homosexual can be used for both sexes, and they mean mostly the same thing; the differences are subtle.

Probably because the word lesbian exists as well, one is slightly more likely to refer to men when using the word gay; but note that it is very often also used for women. When used as a noun, gay seems to refer to men a bit more often than as an adjective, although it is still very often used to refer to both men and women. It rather depends on context. The word homosexual seems to be completely gender-neutral.

Gay is more informal: homosexual is more formal, and so it is more likely to be used in e.g. medicine and biology. As a consequence, using homosexual can sometimes sound a little bit as if you were describing patients, as if it were some mental illness. This effect is not very strong or ubiquitous, but it is sometimes there. It is stronger when used as a noun (an homosexual, homosexuals). Note that gay is currently in the process of being used more in formal contexts too.

The word homosexual is older. Because homosexuality was long treated as a disease or at least as undesirable, some of this old negativity still clings to the word homosexual, especially as a noun. It is as if you were referring to the past, although this effect, too, is not that strong.

I think this is also the reason why the word gay came to be used (around the 1960s? earlier?) to refer to homosexuals: they wanted a more positive-sounding word. You will find that most gays will mainly use gay, except in a scientific context; and even there, homosexual as a noun has become almost impossible. If you were to say homosexuals, and you weren't over 50 or a bit reactionary in general, I would think you were joking.

Gay is perhaps applied more commonly to men (as both definitions state), but isn't exclusive.

As for

When you say “instead of,” it gives me an impression that they are different things

they are different things, but the "things" being discussed in your original quote are words. The statement,

"gay" and "homosexual" are different words

is evidently true. It doesn't imply they have completely different meanings, although they differ in their associations, connotations and tone.

This answer comes to you straight (hah!) from the mouth of a real live lesbian, who lived through the wars of Who Are We, Lesbians or Gay Women?

Women are lesbians. Men are gay men. So there. If you want to use the word "homosexual," then you are a doctor or scientist, or a parent who was raised in the 1930s. Or a member of Congress. Or perhaps a redneck. But then you would say "homo...sexxxxualll" and lick your lips like you were enjoying yourself.

"Homosexual" refers to men or women who bond with members of the same sex.

Anyway, gays took back the word "gay" because it was being used as a slur against them, much like the n-word, much like the word faggot (and if you are unfamiliar with the derivation of the word faggot, I suggest you run to your dictionary right now, since it is no where near as comfortable as the derivation of the word lesbian).

I will force myself to stop here, because I could ramble on forever about the use of these words. Suffice it to say that the word homosexual is an outdated term in all but the most detached scientific writing; that the word lesbian applies strictly to women; and that the word gay can refer to both men and women, but usually refers to men.

terpy

There are some who would argue that "homosexuals" are those who meet the clinical definition of being more attracted to members of the same sex than those of the opposite sex (given that most theories of sexual orientation today indicate that sexual orientation is a continuum, that heterosexuality and homosexuality are more than or less than differences rather than dichotomies).

"Gay" on the other hand designates a cultural distinction. Given the oppression that has led to the "closet," many people who are more homosexual than heterosexual do not and have never participated in the gay lifestyle or culture and may even be largely unaware of it. Those folks may be labeled homosexual but not gay, by some.

Many of my African-American/Black friends make a similar distinction with regard to their culture believing that there are folks of African origin who do not participate in the mainstream American Black culture.