The order of learning words and kanji for Japanese schools and JLPT are completely uncorrelated.
Which is to say that the JLPT doesn't attempt to emulate learning as Japanese people do. So while there is some overlap in the sense that both groups generally follow a principle of going more simple to more complex, what a non-native learning Japanese will find simple and what a Japanese student will find simple can be different.
For a comparison, you can look at this list of JLPT kanji, and this list of Japanese first grade kanji. Both contain about 80 characters, but you can note some differences.
話【わ】(talk), are on the JLPT list and not the Japanese grade one.
犬【いぬ】(dog) are on the Japanese grade one but not the JLPT list.
I know your question is directly about vocab and not kanji, but it was a little easier to locate and look through lists of kanji, and if the kanji the two groups are learning are different, you can be sure the words they use those kanji for are different too.