The Briton started from 20th at the German Grand Prix following a crash in qualifying, but was able to fight his way through the field to finish third, while Rosberg claimed his fourth win of the season.
The 2008 champion told Sky Sports News: "From a personal point of view, I can't be too ecstatic about the race because it has been a tough weekend for me and I've ultimately lost more points to Nico in the championship battle.
"I'm doing absolutely everything I can to get back on level terms with Nico in the title battle. I can't focus more or work harder than I am doing right now.
"This championship is proving a big challenge for me but that's how I love it, and I wouldn't have it any other way."
Hamilton has finished on the podium eight times in 2014, including five race victories.