Neymar is considered a very good player and he was playing at one of the best clubs, with some of the best players, in the world.
Why would he now choose to play for PSG?
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Since I arrived in Europe, the club has always been one of the most competitive and most ambitious. And the biggest challenge, what most motivated me to join my new teammates is to help the club to conquer the titles that their fans want.
Paris Saint-Germain’s ambition attracted me to the club, along with the passion and the energy this brings.
He is interested in their ambitious plans for the future and wants to support major competition victories.
The biggest (and probably only) reason Neymar joined PSG are 36.8 Million Euros per Year + Bonuses. In Barcelona he got "only" 10M €.
For his career it would have been better to stay with FCB. There he would have been able to get even better by playing with and against better footballers than he does now in France.
Obviously it would be hard to discover the true motives behind Neymar decision, but a theory commonly held in media is that in Barcelona Neymar is overshadowed by Lionel Messi (who since 2008 is, together with Cristiano Ronaldo, the only to be considered the best player in each year by FIFA). Comparisons between the two players are common, and even Neymar accepted that Messi is the better one:
Messi is above everyone, there is no use comparing me with him. [...] He is the best player in the world and I always get inspired when I watch good football
There are some indications of this in what some soccer players said:
Brazilian player Dani Alves, now Neymar's teammate in PSG, said that Neymar should step out from Messi shadow:
The thing is, he had to come out of his [Messi] shadow a little. Playing with someone as unique as Leo is the most incredible thing that can happen to you, but there’s always that doubt in your mind if it’s you that really has the quality or if it’s him.
However, Alves also stated that both him and Neymar seek to win the Champions League, so for him this is another reason why Neymar joined PSG (although the chances of winning the Champions League in a team with Messi would theoretically be higher):
It's our main goal. It's what Neymar was thinking when he left Barça and it's what I was thinking when I left Juve. That's what moves us
Also, for Barcelona player Gerard Pique,
Neymar cannot be the protagonist at Barcelona, because the best will always be Messi, always.
Messi's shadow is also mentioned by Thierry Henry, although he stated that he doesn't know if that was the reason why Neymar left Barcelona:
I don't know if Neymar left Barcelona to get out of Messi's shadow. What is true is that all players are in Messi's shadow, and if Neymar doesn't want to be, he should change sport.