A piece of wisdom out of context can also sound wise, but only in an abstract sense. For example, if I were to say here:
"There's a difference between telling a lie and keeping your silence. Sometimes the latter can be useful too. Knowing when to keep your silence and when to share the truth is an art form."*People might nod along and agree with me. But if they didn't ask the question that caused me to say it, don't have a question that needed that particular answer, even if they know what the question is it won't have much meaning to them.
Or they might want to argue with me for how it doesn't apply to whatever is on their own mind, or bring in their own interpretations on what I mean. This can spawn whole separate conversations from the one that originally caused me to say it and have little to do with why I said it.
(Or they might be my little brother, who tries valiantly to find flaws in everything I say and do no matter what it is. :p)
Wise sayings in general - like the kind that sometimes get collected into books, or in my blog footer - are summaries of large collections of life experiences. The people who haven't gone through those life experiences will not understand the intended meaning of the saying that summarizes them. The people who have understand it already.
Wise sayings in context, applied as an answer to a specific example of the problem they're meant to address, can inspire epiphanies and change the course of the questioner's life.
*I said this in reply to an article about "Radical Honesty", via a question on the Whateveresque forum.