I think it all comes down to how they treated the core material, which is Godzilla. And even though Emmerich's movies have plenty of faults all their own (sh!tty characters, same disaster plots, numerous plotholes, stereotypes, etc) they way the '98 movie handled Godzilla is why a lot of people hated it.
The '98 Godzilla wasn't a mutated dinosaur or prehistoric creature, it was a mutated lizard not even the size of your arm. It ate fish, which lizards don't do anyway. It burrowed instead of being an upright, intimidating force. It looks suspiciously like the T-Rex from Jurassic Park, but with lower-quality CGI. Then he breaths fire, which is like a lame half-assed attempt to be like the original.
And a huge factor was that Godzilla always ran away. Shows up, hides underground. Military shows up, runs away. Then at the end it's easily killed by a few missiles fired from jets instead of something hardcore like artillery or something. They literally took the "god" out of it; instead of inspiring fear, people got a T-Rex ripoff that constantly fled from danger and could be easily killed once out in the open. Don't even get me started on the velociraptor chase scene.
The 2014 Godzilla on the other hand was a beast that truly lived up to the name Godzilla. Aside from being the largest Godzilla incarnation so far, we see him arise from the oceans, and whenever he's on screen (and even when he isn't) you can sense his presence. The tides shift, boats are shoved out of the way, the ground trembles, debris and dust fly everywhere. Even his roar feels like it has force behind it. Godzilla is a natural disaster, and his atomic breath reminds us he's from a radioactive age and doesn't belong in the modern world.
Plus this Godzilla fights other monsters. We can't tell if he's fighting for us or if he's merely fighting the MUTOs because they're the next biggest kids on the block. The '98 Godzilla was like a fluke, "Whelp he damaged NYC, we killed him after a day or two, now it's back to normal." Because of the MUTOs, you can't help but feel like there's even more giant monsters laying dormant that aren't seen in the 2014 film, which adds a certain level of mystery and minute horror to the film.
Even Pacific Rim had it right with the Kaijus. They were wholly original creations but designed to elicit fear through their size and actions, much the same way the 2014 Godzilla was.
Personally I loved the 2014 Godzilla film and can't wait for Godzilla 2 with Rodan, Mothra and Ghidorah added to the mix. Especially since they've already announced that Godzilla & the monsters will get more screen time, which was really my only gripe with the 2014 movie.
Can't be too careful with all those weirdos running around.