Tearing up a little bit is what Naruto did, he barely shed a single tear. Hinata on the other hand had tears rolling down her cheeks and her body was even shaking. We even see in 699/700 that she cried for him again and that over a decade later, despite having a family to look after, still visits his grave.
Still.... a lot of people seem to think that Hinata is obligated to care for Neji simply because of their blood relation. I mean, I understand the two care for each other, but people make it out to be like Hinata isn’t ALLOWED to put Friends over her Blood Relatives, and this makes 0 sense. Blood relations is not something that determines the deepness of a bond you have with others.
But honestly, these two moments aren’t really a good comparison. Because in Naruto’s case, we have Naruto in the process of dying, and Hinata is unaware of his condition, while in Neji’s case, Neji has already said there is nothing that can be done to save him, and she is aware of this.
Both of these moments were devastating for Hinata, and even if she did care more for Naruto than Neji (even though I don’t see how you’d even be able to measure the “careness level”), this doesn’t in no way devalue the Care she showed when Neji died.
But hey; in the end, no one *****es about Shikamaru crying over Asuma but not Shikaku and no one *****es about Naruto crying his eyes out when Gaara and Jiriaya died but barely shedding a tear when Neji died. Oh my! What is this I see? I double standard, perhaps?