Old = having been alive (or existing) for a long time. My grandmother is an old woman. This book is old -- it was published in the twentieth century.
Old fashioned = in style or popular a long time ago, but now out of style. My grandmother uses old-fashioned slang rather than what kids use today. Clothes from 1920 are old-fashioned. Sometimes people used old-fashioned as a synonym for modest, especially if they're talking about clothes or morals. Other times, people use old-fashioned as a kind of insult, implying that the person is refusing to adapt to modern ways.
I hope that helps!