Seems like after some looking around this is what I make of it (don't completely take my word for it though):
Både is always used with og. It can reference two or more things.
Jeg er både træt og tørstig.
Både Jens, Erik og Else er der.
Begge references two things that don't have to be listed.
Begge drengene har jakkesæt på.
Min tante kan godt lide dem begge.
Again, take this with a grain of salt as this is just what I've found after doing only a few minutes' research on it, but this is what I make of it.
Whoops, just realized the document above is about Norwegian grammar. However, my guess is that the same rules apply in Danish. :-)
Hm I could use some clarification as well. http://ordnet.dk/ddo/ordbog?query=begge&search=Søg Ordnet says both one and the other of two.
"trætte" is plural form of tired. "De er trætte". Although "tretten" (13) is pronounced almost the same. Maybe thinking of 'tretten' as a word you say fast and 'tredive' as a word you say slowly helps?