Pacific Rim is the superior film. Godzilla feels like it's tugging at too many strings hoping one of them will pay off in connecting with an audience member, whereas Pacific Rim seems like a much more cohesive and honed experience.
If these two movies were RPG characters Pacific Rim would be heavily spec'd in Strength or Dexterity while Godzilla would have an even balance of attributes. The effect is that Godzilla probably has a much wider appeal but is less enjoyable overall, and Pacific Rim probably alienated a lot of people outright but those who were accepting of it would enjoy it all the more.
While the monster parts in Godzilla where pretty awesome, the human side was generic, long and boring. And, while Pacific Rim started and ended with Robots vs Monsters action, Godzilla has 2 hours of "look at the american hero running around while he tries to find his family... and here is some footage of a giant fin, I think..."
I choose Pacific Rim, however, both are great movies.
Both are "Kaiju" movies so you have to weigh in their monster battles heavily. This is where Pacific Rim clearly wins as it has more monsters and mecha, more battles, more action, and better "effects". Godzilla has better acting, the better build up, and I would say the better individual scenes. In the end it is about the monsters though! That being said Godzilla is still an awesome movie and probably going to be one the bigger ones of the summer, everyone who just likes fun/good movies should check it out.