I love both games, so let me give a rundown from my point of view:

Civilization is more "gamey". It has a boardgame feel (but with much more depth than most boardgames). It has mechanics that are simple to grasp, but that lead to very deep strategical and tactical decisions. That's the main beauty of the Civilization series for me - they are very easy to pick up and get started, but are still quite deep and take a long while to truly master.

Europa Universalis has a different style. There are no set objectives, there's no "victory". It's sort of an "Alternative Story Simulator". EU4 is probably the most "gamey" entry in the EU series so far, and it has pretty good gameplay, indeed. You'll be faced with decisions that have a lasting impact on your game. It's a much more complex game than Civilization, and it's not as friendly to get into, but it's very satisfying to play, especially if you happen to enjoy history and geography (as EU is set on actual historical time setting and real world map, as opposed to Civ).

As a side note, despite EU4 being real-time and Civ5 being turn-based, I feel that EU is more "slow paced" than Civ. You'll have a few years of "downtime" at times, when you do few things besides waiting.
At the end of the day, both are great, and it's really down to which genre you want more.

On a last point, if you're going to pick up Civ5, you'll need the DLCs. The two major expansions (Gods & Kings and Brave New World), at least. They add a lot to the game and greatly improve it. For EU4, you don't really need the DLCs, they're quite optional.

The most important thing you need to understand IF you want to compare this games is that in Civ V, you have a goal in the game. You can actually WIN a playthrough.
In EU IV, its not about wining, but creating your own goals and achieve those. (yes, you have score, but that is not important)