A little disclaimer to begin. I don't use Chrome. Despite the fact that it is damn fast, reliable, and tends to pick up new web features faster than any other browser, I prefer to limit my dependence on Google and so I will stick to the more open Firefox for my web needs.
What's even better about this arms race compared to the one in the 90's is that Mozilla and Google are actually cooperating to make sure that things are standardized. None of the old problems where Netscape would use one name but IE would use a different one - no, now we know that after the APIs become stable there will be efforts to standardize them across the more modern browsers.
One thing I've deliberately left out of the discussion is Internet Explorer. It continues to be developed, but as is typical of the IE team there doesn't seem to be a whole lot of effort to make it compatible with the things that Google and Mozilla are building. I think that the web would be better off if developers showed a bit of defiance and just decided to ignore Internet Explorer: focus on building amazing applications for the browsers that are actually pleasant to use. The users of those browsers will be much happier that way, and more people would migrate to those browsers when they see that they can have a much richer web experience that way. To quote some advice I've heard from various VCs and startups, "focus on making a few people very happy than trying to get a lot of ambivalent users."