Are the first few grades worth of kyōiku kanji exactly the same kanji that are on level 4 of the JLPT?

No.

A lot of people try to learn Kanji for words they don't know and end up forgetting them, because they never use them. So, its really better to try to learn the kanji for words that you know. That way you will always have a chance to use them and you won't be trying to learn an abstract symbol for something that means nothing to you. I always encourage my students (I teach a Japanese as a volunteer) to think of kanji as the spelling of the word. When you learn an English word, you also have to learn how to spell it right? So if you write "difekult" instead of "difficult" no one will know what you mean/won't take you seriously. The same thing is true of kanji. You can write むずかしい and people will know what it means. But you will be taken more seriously and people will think your Japanese is better if you write 難しい. It is the correct "spelling" of the word. I will tell you, this is just my personal take on things. Most Japanese teaching material is written by native speakers. Their instinct is to do it the way they were taught. So, they tend to teach it by grade levels. However, a first grader in Japan already can speak Japanese. They are just learning how to write words they already know. For that reason, I recommend Japanese learners to do the same thing. Learn to write words you know.