I play both. I have almost 300 hours of Civ5 and 600 hours of Shogun 2.
Interface & Micromanagement
Shogun 2 has similarities in being very slick and streamlined in gameplay and interface (both games have areas that are lacking badly. Using the interface to find/control air power in Civ5 is a chaotic confusing cumbersome mess. Shogun 2's ship building, movement and ceued end-of-turn movement actions are all a pain to deal with. Civ5 is a chaotic mess in late game with many cities and units that cannot be renamed or grouped into armies (no stacking allowed etc). Shogun 2 is smooth sailing in comparison.
Shogun 2 requires only a fraction of the micromanagement of Civ5. The economy is much more transparent and easy to understand. There is no "CHOOSE PRODUCTION" or "UNIT NEEDS ORDERS" popping up every turn. It is instead 'on demand' - you go and recruit units somewhere if you need to. Trading is also cleverly done in Shogun 2, with mininal interaction needed from the player. Civ5's constant requirement to renew all types of trade, and to do it manually for each resource, open borders, pacts... A pain. Denouncements and Declarations of Friendship also constantly need to be renewed.
During battles it is a different story, where one micromanages a formation of units. Not as intensely as Starcraft 2 though, it is more peaceful.
Both games have problems with transparency. They hide mechanics and important information from the player, such as how many turns are left for this deal, and what are the consequences for breaking this deal within this amount of time, etc.
The advisors of Civ5 are a nice idea but they are flooded with excessive unwanted information, drowning out the few things that would be of value. Often the info of the military advisor is the same thing stated in different words, over and over. I don't know why they didn't make the advisors appear (unprompted) to tell different things the player wants to hear about.
General
Shogun 2 consists of two 'games' in one in a sense. A campaign strategy (like civ5) mode and a battle (Rea Time Strategy) mode. I thought such a combo would cause a loss of focus and quality, with one or both being shoddy. But rather the opposite, both are of exceedingly high quality. They tie into each other in a brilliant way. What happens on each mode carries into the other seamlessly. Some people play only head to head multiplayer battles, forgoing the campaign mode. There is a cooperative multiplayer campaign mode, as well as an adversial one (which is like the normal singleplayer campaign, but with two humans instead of one). Many clever ways to have humans replace the AI during battles during campaigns. There is the 'drop-in' mode where one can join others' campaigns as the AI during a battle, and vice versa.
Shogun 2 has a narrow focus (location, time period, units, objective) while civ5 is more epic. Both are a bit like "a really good board game". Shogun 2 almost looks like it on its campaign map.
Shogun 2 of course has the same map, every time. Same starting year. Same conditions. But the geopolitical & clan difference makes for very different game experiences depending on what clan one chooses. Every province is already laid out and cannot be changed. It belongs to a clan or rebels. Provinces have a single 'castle town' (like a city) each that cannot be destroyed, only change owner (and developed, of course). Provinces also have variying levels of fertility for their crops (which affects how weathy it is) and may have a harbour or not. Provinces have different types of specialties, which is like having bonus buildings or wonders in Civ5.
Shogun 2 has religions. More elaborate and detailed than Civ4, yet easier to manage and not as 'religion determines if we can be friends or enemies' as it was in Civ4. It plays a major part in diplomacy, economically and for the military. It is intricate and great fun.
Diplomacy
The diplomacy in Shogun 2 is exceptionally well thought out and implemented. It is vivid, alive, entertaining, mostly logical and stimulating. Almost every turn it is worthwhile to go 'read the news' of the latest happenings in Japan, and see how the clan relationships are developing. Cunning and forethought highly rewarded. Much manipulation to be made, but not excessive. Civ5's diplomacy, while easy to understand (mostly), plays only a minor part and there is little players can do (and it gets worse the later in the game one gets).
Shogun 2 also has minors and majors. But the difference between them is nothing like Civ5. All playable clans are majors. They have unique units & clan traits. Minor clans often become super powers. The interaction between clans makes no distinction between them.
Misc
Shogun 2 has loads of beautifully rendered movies playing now and then, nicely narrated. When using ninja agents on missions, it plays clips lasting about a minute, detailing the ninja's mission. Different settings, characters and methods. He can succeed or fail at various stages, and they rendered movie clips for each such possibility. There's tons of them, and even after a long, long time of gaming, the movies are still refreshing, exciting and often pretty funny (example: Samurai retainer sleeping in a field grabs the ninja's leg and dreams of cuddling with a geisha. Ninja tries to get loose and then it may or may not succeed...).
Bugs
Both games suffer from bugs. Civ5 is the worse offender with multiplayer being, well, a joke, especially when AI civs are involved. Latest patch supposed to fix the AI for MP, but it remains to be seen. The courthouse bug in Civ5 (old yet alive) is taking away much enjoyment from my games (the bug is that courhouses remove not only the occupied penalties, but also the -3 unhappiness that a city should generate. Annexing thus generates VERY HAPPY cities, much happier than one can build oneself, while puppeting generates more unhappiness, completely opposite to game description and intent).
Performance
Performance wise, Shogun 2 runs better, but has extremely long load times. On my high end computer, Civ5 scrolls choppily along the map no matter what settings, while zooming close to the ground or not moving the view is smooth even at the highest of high settings.
Summary
I like them both. Shogun 2 is the better game with more replayability, variation in tactics and strategy (unexpected, due to the narrower focus) and overall more stimulating and satisfying. For me it is the finest game ever made (I have a long gaming career - since 1985. I own the original Civ1 on the Amiga, with the nice big black manual and everything). But Civ5 is a terrific game as well. If the MP was fixed up to SP standard and the courthouse bug fixed, it would surpass Civ4 easily and equal Shogun 2.