I noticed that if I make this my default browser, that certain internal Android URLs cannot be resolved. For example, clicking on a market:// link on a web page causes an error; and if Android attempts to open a myapp:// link (e.g., when you do a Twitter auth and it tries to redirect you back to your app), you get an error. This made it impossible, for example, for me to successfully authenticate my Twitter account in a Twitter client I installed. I had to temporarily reset the default browser back to stock, do the authentication, and then reset the default browser again back to MIUI. Not that I do these activities regularly, but it is frustrating that the MIUI browser is lacking these core implements of the stock browser.
Also, I found that if you remove the stock browser, Google Search and Voice Search break (tied into the stock browser app apparently). So even if MIUI is your default browser and you'd like it to be your only installed browser, you can't because then you'd lose Search.
Other than that, I do really love this browser!