When I look it up, Brüderchen seems to just mean brother. Then, what's the difference between it and Bruder? Why is it spelled with an umlaut and "chen" on the end?
"Chen" is a diminutive ending. If "-chen" is added to a word the word becomes neuter and the vowel very often takes an umlaut.
- (der Bruder) das Brüderchen - little brother
- (Vogel) Vögelchen - little bird
- (Haus) Häuschen - hut, cabin
- (Brot) Brötchen - bread roll
Another similar diminutive ending is "-lein" (der Bruder - das Brüderlein).
I don't really like that question, because that could have been answered with a little bit more research
Why is it spelled with an umlaut and "chen" on the end?
As said in the other answers it's a diminuation of the term Bruder.
Then, what's the difference between it
The semantical nuance for doing this is from lovingely to discriminating, and totally depends on the concrete context of those siblings talking.
The grammatical building of the Umlaut combined with -chen (or -lein) is quite regular though.