I think you could debate either way.
On one hand Bergeron is comparatively overrated and Kopitar underrated due to both markets and countries. However, fortunately, boston fans are very flattering of Kopitar and many put them on the same level.
I think Bergeron is an underrated complementary player--he's great offensively with his elite wingers, and obviously played like a madman next to Crosby for Canada. I'm not convinced Kopitar can do similar.
On the flip side, Kopitar I personally think is better at being 'the guy,' I'm not sure Bergeron could carry a team in similar manner, and I don't mean that as in insult to Patrice but a compliment to Anze.
I do think, ultimately, Bergeron is the better strictly defensive player, Anze the better offensive player, and personally I believe Anze to be the better all-around player, but I'm not opposed to the opposite argument, I think it's very close or at the least VERY debateable. I DO think if Bergeron were not playing Anze would have several more Selkes and isntead of talking about Bergeron as possibly the best defensive forward ever they're talking about Kopitar being the best defensively of this era nad in the end that's the real crime here, Bergeron's shadow.