The main difference is that Brikka has a pressure-release valve, which allows for a slow, even steam release. This results in a rich crema layer and less chance of a bitter, burnt taste.
Brikkas are also a bit more expensive than regular Moka pots. Also, due to the higher pressure, Brikka pots are faster in making your coffee. Though, I don't think it makes much difference, as you'll only save about 30 seconds.
Another essential thing to notice when choosing the best Moka pot for you is that classic Moka pots can be electrical or compatible with many stove types. On the other hand, Brikkas, at least for the time being, is only compatible with gas stoves.
Both Moka and Brikka pots produce a robust and flavorful espresso. So, it really comes down to personal preference. If you want a richer, crema-topped espresso with less chance of a bitter taste, then go for a Brikka pot. If you don't mind a slightly longer brewing time and want to save some money, then a classic Moka pot will do just fine.
In my opinion - If you have a gas stove at home - go with the Brikka. If you are looking for more versatility in where you can use your Moka pot - go with a classic one.