On the new Calendar in Mountain Lion I just noticed that there are not two ways, but three ways to interact with the reminder pop-up window - you can click on "Close" or "Snooze" buttons, or click anywhere else other than those two buttons and it brings up that event in Calendar ready to modify the alert that caused the pop-up or even add another alert.
I agree with you it seems not as useful as iCal on Lion and previous versions of the calendar program, but in some ways it's more.
Another new feature is that in the Preferences there is a new pane that lets you set the default alert pre-warning for each account for three class of events - normal events, all-day events and birthday events.
It's one more click than before (maybe more depending on what you want to do), but the added advantage of the default alert is something I'm liking more and more. You can even set it to None if you don't want a default - I think this was the way iCal worked before when creating new events.
So what I do when the pop-up appears is click in the main body of the pop-up to bring up Calendar with that event already opened, then set a second alert that is the period of time I want to be reminded next. So now you can specify any sort of reminder rather than the fixed list of choices that iCal gave you before - which is something I've always wanted an easy way to do.
This new behaviour seems to be a compromise of giving you the ability to specify any length of reminder combined with the ability to immediatly dismiss the event (by clicking "Close") or snooze for a few minutes (by clicking "Snooze"). I still hope that Apple lets you specify the period of the "snooze" if you want - perhaps with a "defaults write com.apple.Calendar DefaultSnoozePeriod 600" Terminal window command or something like that (600 would specify 10 minutes in this example).
I think a better solution would have been to have the click on the "Snooze" button bring up a list of reminder periods like it did before with the added option of being able to modify that list according to your own desires by clicking somewhere other than Snooze or Close.
Anyway, after using Calendar for a while I'm getting to where I'm not as upset as I first was when it appeared that we had completely lost the ability to as easily modify/create a new alert if desired as iCal had previously let us.
Try it out and see what you think.
-Bob