How come that the manga of Boruto: Naruto Next Generation is different than the anime?
Short answer: the manga is one arc ahead of the anime.
Long answer: Actually that's a little bit more complicated than "the manga and the anime are different". They are actually mixed up and follow different timelines. Also, there isn't one manga or one anime but actually two different mangas, a film and an anime entering his second arc. Let's get right into it :
Note: all release dates are Japanese dates.
The story of Boruto starts on August 4, 2015 with Naruto Gaiden: The Seventh Hokage and the Scarlet Spring, an independent book published in the Weekly Shōnen Jump. Along with this book, a movie called Boruto: Naruto the Movie is released on August 7, 2015.
The character of Boruto is then added to the original manga on October 6, 2015 in the 700th and final chapter of Naruto, when he becomes Hokage. An OVA of two episodes called The Day Naruto Became Hokage is also released on July 6, 2016.
Then, a whole new series called Boruto: Naruto Next Generations started on May 9, 2016 for the manga and April 5, 2017 for the anime.
To summarize, we have 1 short book, 1 chapter, 1 manga, 1 anime, 2 OAV episodes and 1 film dedicated to Boruto. That's a lot, isn't it?
So what about the history?
To follow the timeline, it starts with, of course, the 700th chapter of Naruto and the OAV "The Day Naruto Became Hokage" which basically recount the day when Naruto became Hokage and show us the Naruto's main characters' children.
Then, we have the first arc (18 episodes) of "Boruto: Naruto Next Generations" the anime. Boruto and his friends are yet applicant ninjas at the Ninja Academy. The anime emphasizes Boruto's journey BEFORE the Chūnin Exam.
Then, we have a special arc dedicated to Uchiha Family with the short book "Naruto Gaiden: The Seventh Hokage and the Scarlet Spring" and the second arc of "Boruto: Naruto Next Generations" the anime.
Finally, comes "Boruto: Naruto the Movie" that takes place DURING the Chūnin Exam and "Boruto: Naruto Next Generations" the manga which recounts the film's events but also emphasizes Boruto's life AFTER the Chūnin Exam.
That is a little complicated so I hope it was clear.
Although I agree with everything in Nevios' answer, from another lens, I would also say that the manga and anime are different because the manga only comes out once per month and the anime would be ahead of the manga which is a no go because the manga is typically looked at as "canon" or "the actual story line" while animes have the luxury of having "fillers" or "story lines separate from, and in some cases, alternative to the actual story itself.
I believe the creators of Boruto made a brilliant decision to go back and fill in the gaps between Boruto starting his ninja journey and the Chunin Exam. By doing so, they brought back some of the nostalgia of the Naruto series while also allowing us to understand Boruto beyond what we saw in the movie which was a character that was really bratty but redeemed himself at the end.
Through the anime, you get to see that Boruto has a strong moral compass and that he isn't just looking for the easy way out. Conversely, unlike Naruto, Boruto has had everything he would have ever needed from jump. He doesn't know struggle like Naruto or Sasuke did. It's a poetic commentary on our technological generation vs the generation before. Boruto has a mindset of "why do things the hard way when I can get the same result with an easier path" while his elders are trying to teach him that there is something very important about going through a process.