They are generally used interchangeably in common english. It is similar to the difference between due to and because of. Though you can switch them you would ascribe something to a noun. Example: I would ascribe the symptoms to allergies. Attribute is similar to because of. You use because of a verb. Example: I would attribute the symptoms to running in the cold.
Attribute can also mean a characteristic if you are using the word differently!
Also, in comparison to ascribe... attribute is more commonly used in english when discussing the causes of things.