When you do something wisely, you act in a thoughtful way, based on your own knowledge. Did you use your money wisely? Or did you spend it all on candy?
Use this adverb for things people do that are (or appear to be) based on good judgment and experience. Smile wisely at your friend and they might think you know what you're talking about when you describe the UFO that landed in your back yard last night. But if they wisely ask you more questions and start to look skeptical, don't be surprised!