The examples provided by my preceedesors are useful, but the distinction between "znać" and "wiedzieć" is not purely about the topic, but can also be influenced by the way it's expressed. Let's see the way we know a certain method:

"Znam język angielski" – "I know English" (much more common)
"Wiem, jak mówić po angielsku" – "I know how to speak English" (mostly theoretical, but still valid)

Forgive me if I am about to write something stupid, as I'm writing this without any research into the topic, but I feel that when you can (in Polish) immediately follow the verb for "know" with its object, you use "znać". If it would require a preposition or something, then use "wiedzieć".

"Znać" can refer to people, song, movies you know, while "wiedzieć" is more about the knowledge. Examples: Znam ją! - I know her! Znam tę piosenkę. - I know this song. Wiem, że nie przyjdziecie. - I know that you won't come. Oni wiedzą o nas! - They know about us!

Since you are learning German, the best explanation of the difference would be:

• znać = kennen

• wiedzieć = wissen

English doesn't really distinguish knowing something from having knowledge about something, so to English language both translate to the same thing. ;-)

As a native speaker (I am Lithuanian) sometimes I'm stuck between them too and I use "znam" in context as I would use it when speaking Russian. But I feel the difference, like, znam you use more when you "know" someone (a person), and wiem when you "know" something (any fact, etc.).

"znam" - I know [someone, something] "wiem" - I know [it, about something] Znam go. - I know him. Znam ten film. - I know this film/movie. Wiem. - I know it. Wiem o tym. - I know about it. I hope I helped you ;)