You're misunderstanding the point. Actions speak louder than words mean that your actions bring out your character. Pen is mightier than a sword means violence will not get you anywhere. The two knobs of go together.
Actions speak louder than words because you can say all you want but in the end how you act is how you are going to show people who you truly are. Written word is man's greatest achievement because with books and written things, we can teach, learn and succeed.
Pen can keep a person's reputation alive for generations, such as Alexander the Great. We wouldn't know who he was today without writing, although he ruled most of the world and was a great warrior. Writing is an action. Talking about writing is just words.
Action shows true intent, writing gives the author time to think and look over what they have said. The spoken word is mixed and can be said out of anger, the written word can carry the weight of the sword. The truth is the difference, they both speak of actions that have cause and effect. The difference is the situation.