Kakashi spent years and years grieving on Obito's and Rin's death remember? When we first see Kakashi, he was visiting the KIA stone to pay his respects to his fallen comrades. The fillers even give a look at Kakashi's life after their deaths and how he distance himself from everyone and only Gai, Minato, Hiruzen tried to get him to deal with their deaths.
So why is Naruto having a more powerful reaction? Well he's not, Kakashi had years upon years of dealing with Obito's and Rin's death and Kakashi was looking down just like Naruto when Obito was dying
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. Kakashi was influenced by Obito after his death is him teaching team 7 about teamwork is stems from Obito since he now will put his comrades live over the mission. He also took on some of Obito's traits like showing up late. Obito is also why Kakashi would say he would never let his comrades die, he tells Naruto this when they were fighting Obito.
