I always preferred national team competitions. Maybe it's because I used to watch those more (yes, some years ago I had no international channels and Portuguese TV sucks), but it's also the amount of stars that gather, the all or nothing games, the chance to see players you often don't, etc..
Club. I feel much more passionately about my favorite club than my country, to be honest. Sure, I always root for my country to win and there's a sense of national pride, but if you constantly follow your favorite team through ups and downs, failures and successes, you bond to the team much more than a national team which you watch just a few times a year.
I enjoy watching national play too, though, since as you said you learn about the best players from each country and there are some great teams.
National team for sure. There are many reasons for that. First of all it is a national pride. There are no foreign players in the team. The atmosphere during NT competitions is different in the whole country. Also we are always aiming for the top places and most of our players play abroad so there aren't many chances to see them play in the club season.
Club competitions of course. Especially the local derbies or the traditional European derbies such as PAO-Maccabi for example. Nothing gets the adrenaline flowing like such games do.
NT means little to me.