I was talking about the self same thing yesterday with a friend, and we both agreed that, like you, the Champions League is a greater competition. You can be sure that you're turning up to a game of Europe's elite.
It has a variety of world players, just like the World Cup, yet, I feel the passion is greater for club teams.
Its great going to a Champions League game on an autumn night. The music as the teams line up, the stunning vocals of the away fans, barely matched in league games. But, more importantly, the competition is spread over the season. The time period builds anticipation and allows time to savour games better.