В чем разница между bevorzuge и lieber ?Пожалуйста, приведите примеры.

"lieber" is mostly used as the English "rather (than)", it often implies a choice between two (or more) things/options.
"bevorzugen" is "to prefer", it expresses that you favour something

I guess in most situations, you can use both, since they're equivalent in meaning. In official situations, I'd always use "bevorzugen", since "lieber" sounds a little too personal/familiar in my opinion.

Ich bevorzuge es, draußen zu laufen.
Ich laufe lieber draußen (als drinnen).

While the 1st sentence is a plain statement of a fact, the second one implies at least some context, namely that you have either been asked "Would you rather run outside or inside", or that the other option is obvious.
This difference is pretty subtle though, and how it is perceived might also depend on the person you're talking to as well as the actual situation.