by grant5
The BGG glossary is probably as good a place as any to start from. Based on the definitions they give for broken and solved, they are definitely related and I admit they sound similar.To me, broken means that there is something in the game that a player can exploit to guarantee victory. The Halifax Hammer and Caverna's infinite ore combo qualify as examples of this for me.
Solved means that the outcome is fixed before the game even begins (or maybe after one or two moves). The optimal outcome can be mathed out by a computer.
And there are very few BGG ranked games out there that should have either of these terms applied to them. Most often, when people use either of these terms, what they really mean is that the game has a dominant strategy.
https://boardgamegeek.com/wiki/page/glossary#solvable
https://boardgamegeek.com/wiki/page/glossary#broken
