No, they are not the same, although they have similarities. If you influence something, you have some effect on it. If you influence a person, you have some effect on that person's thoughts, feelings, or actions. If you persuade a person, you get that person to do something or believe something. This is something you do intentionally--it does not just happen.
If you persuade someone to do or believe something, then you have had an effect on that person, so you have also influenced him. For example, if I persuade you to eat at a certain restaurant, I have also influenced you.
But if you influence someone, you are not necessarily persuading him. For example, if you are around people who use a lot of slang words, you might start using them too. Their behavior has influenced you, but no one has deliberately tried to get you to speak the way they do. So you have been influenced, but not persuaded.
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Influence generally refers to affecting someone's opinions or actions, while persuade specifically implies convincing or inducing someone to do something. While they are closely related, persuade typically involves a more deliberate attempt to change someone's mind or behavior.
Words that mean the same thing are called synonyms.
The term for words from different languages that look similar and mean the same thing is "cognates."
Two consecutive words in the one sentence that mean the same thing are called "redundant expressions."
the same thing mean "la même chose" in French
Other words that mean the same as "different" include diverse, varied, and distinct.