Transsexual is a bit more simple because it's assumed that there are only two sexes, the one you're born and the other. If you're born male but you feel you're female, in case you decide to change from the outside then you become transsexual (trans + the sex you identify with), and with nowadays techniques you can become mostly undistinguishable with other women. The opposite process is exactly the same.
Transgender came years later when it was being discussed if there are more than two sexes and you can have any of them (sex is always a biological term) but a different gender (identity) which doesn't equal to 1:1. For instance a male is not necessarely a man, you can be what's understood traditionally as a Western man (brawny, heterosexual, following a dress code and way of life, etc.) but you can feel that your identity is other by changing any of the patterns. For instance, wearing high heels, a skirt, a cleavage suit, a wig and lipstick doesn't make you less male than your neighbour, but your gender will not be the same than the previously described. Without changing your physique or clothing you can autodefine yourself as a woman (same again, there is more than one type of), as genderless, genderfluid, non-binary (or the third gender in Germany), etc. The palette of colours is vast. So even two CIS people can differ among themselves when it comes to gender.
Outside all this, you can also simply not identify yourself with any. Some people care about it and they ask to be legally changed to male or female (man or woman in legal terms in those countries that allow it) if their ID says otherwise. When legal matters come around there's a new debate, should a trans woman and a woman or a trans man and a man have the same rights, even they're biologically different? This has not been resolved yet and it causes a lot of headaches to legislators, since working females may need a day or two off every month if they have an uterus disease that affects menstruation but a trans woman not, same for pregnancy or when it comes to jailing (and of course when countries don't allow marriage between same sex couples).
While IRL there is no consensus, in otaku media there's just a few gender topics and most of the times treated superficially, summarised with "can a boy like cute things and a girl manly things?". Seemingly with sexuality, there's usually a double-faced character (they only show who they are to those they trust or don't care at all because nobody questions them) and it's always summarised as "I am who I am". And in their best attempts to overcome this, they need to mix romance with it, so very very rarely it's something different from yaoi/yuri/bi with cloathing components.
