I've just learned that there are long and short words for the same thing, like больной & болен. What's the difference (apart from the obvious ones) and which of used when?

See a translation

this is a different part of speech, but the meaning is the same

Это одна часть речи и смысл у них одинаковый, во втором случаем краткая форма слова 'больной'

All adjective in Russian have short and long form: добрый/добр, добрая/добра, доброе/добро, добрые/добры. And others same way. It's means almost same things.

Short form adjectives have wisdom shade.

It's hard to explain... in fact, long form represents an absolute nature, which is not associated with a particular situation, while short form represents relational nature in the particular situation:
1)Потолок низкий (low ceiling) - it's a fact
Потолок низок (the ceiling is low) - for the furniture in the room, for example
2) Ботинки маленькие (small shoes) - it's baby size
Ботинки малы (the shoes are small) - they are small for me

Oh, and another difference is that long form represents a permanent nature and short form - temporary nature:
Он очень добрый (he is very kind) - he is a good man and it is a fact.
Он так добр! (He is so kind!) - it means that he has done something kind and I said about it. He is kind right now. We don't know what a person he is.

But there is practically no difference if we haven't the context. We may use both forms if we don't elaborate the details. For example, we can't say "Эти ботинки мне МАЛЕНЬКИЕ", we have to say "эти ботинки мне МАЛЫ"
//but we can say "эти ботинки для меня слишком маленькие" (these shoes are very small for me). The power of Russian xD//

And the last difference is in grammar. Short forms of adjective or participle may act as a verb and long forms as a noun:
Раненый лежал неподвижно (the patient was laying without moves)
Он был ранен (he was wounded)

I hope you will understand something, I really tried to explain it. And sorry for my English :)

Before a noun only long forms are used. (Больной человек)
As a predicate long and short forms can be used (Человек болен, Человек больной) There is subtle difference in use and meaning too.

больной - полное прилагательное, оно обычно является определением (подчёркивается волнистой линией), если в предложении есть глагол или другое сказуемое
болен - краткое прилагательное и оно всегда является сказуемым (подчёркивается двумя прямыми чертами)

"больной" может употребляться как существительное. оно похоже на слова "учительская", "столовая", "детская"

@AndreyW :D