The difference is, from my perspective, Despacio has a wider range of meanings. Road signs in LaAm countries I've been to say 'Despacio' as a command whereas you'd never use lentamente that way. You can pretty much use despacio/despaciamente/lentamente almost interchangeably in most other situationis.
In addition to markgjensen's link, there's this one, which I use to tell the difference between para and por. You might not need it, but some people might find it useful! http://www.dummies.com/languages/spanish/differentiating-between-por-and-para-in-spanish/