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Rome Total War: I'm not gonna criticize it due to the time it was made. But I personally recommend you play it only with mods. As you can easily cheese the AI out of practically all their settlements if you abuse the vanilla diplomacy enough. Units rout quickly, and the AI is completely dumb on the battlefield. So yeah, mods or bust.
In terms of gameplay/empire management. You only have cities, as castles are introduced in Medieval 2. Diplomacy is a real big pain in the butt to do. You have to send a diplomat all the way to a faction's army/holding in order to negotiate with them. This means spending 10+ turns sending one diplomat from Rome to Egypt if I wanted trade with them. Your cities will increase in population, and the next city type will unlock more buildings/units. So that's essentially your technology in the game.
Medieval 2 Total War + Kingdoms: Pretty near, this is another you must play with mods. As the diplomacy is very hard, if not impossible to keep at higher difficulties. Or being paranoid when Sicily is going to send an invasion force all the way from Italy to England just because. AI is once again completely dumb in this as well. All higher difficulties do is make them spam units much faster and that's practically it. Medieval 2 has the best mods though, especially Stainless Steel if you're looking for hardcore realism.
In terms of gameplay/empire management. You have two options for cities. Cities for economy, or Castles for military. This makes you adapt the playstyle of changing your inner holdings to cities to fund your military conquest on the borders. The AI will always attack your border cities, so you will know where they are coming from. Now if we talk about Kingdoms. This is by far the BEST expansion I've ever played. This introduces 4 entirely new scenarios, the Teutonic Knights, the Americas, the Crusades and Britannia. Each with 5+ playable factions and little quirks.
Empire Total War: Eh, this one is pretty alright. It's not the best in the world, but when you have the Iroquois Confederacy trading a territory in the Americas with Venice for Greece. You know the AI is stupid. On the battlefield they show their extremely incompetence, and on higher difficulties only win because of their ridiculous stat boosts. So expect to have your line infantry struggle against militia units on the high. This used to have a coop multiplayer, but it was a beta for Napoleon and was never made publicly available. They change up the formula slightly in terms of economics for this game. In Medieval and Rome, you primarily relied on population growth to get your money rolling. In this, they introduced the town growth system. Where if you have lower taxes, your town wealth will grow really slowly. This is primarily used for end-game focus. Empire is an attempt to add a story campaign which I quite enjoy. It takes after the forming of the colonies to the War of Independence.
The early-game cannons are absolutely atrocious. Never bother with them unless you're the expert of cannons Napoleon himself. The fun comes in with motors and rockets. They also changed up how technology works in this game compared to the previous titles. As you have to actually research them, instead of "Reach population x to upgrade city/castle x to x tier". Which I quite enjoy. Also going back to AI for a second. Expect to get REALLY annoyed when the AI you're at war with keeps sending 1 stacks just to harass your economy just to be annoying. I really do enjoy the theaters in this game. Instead of just Europe, they actually let you fight for Europe, Americas and India. Funnily enough they were supposed to make the Mughal Empire a playable faction. Cause if you edit game files to make them playable, they actually did have a starting introduction but not voiced. So don't know where that went CA. Despite the obvious flaws in AI, it's a decently done game that mods can help. But not fix the core issues, only paint over it.
At least I don't have to get a diplomat agent and walk him across miles just to talk to someone now.
Napoleon Total War: If you aren't already attracted to the concept of the Napoleonic Wars. Then it's essentially a more focused Empire Total War that introduces SEASONS. Napoleon lets you play as Great Britain, Prussia, Russia, Austria and France (if you complete the Napoleon story campaign). There's really not much else to say that I haven't already said in Empire except cannons are actually good in this game.
Shogun 2 Total War + Rise of the Samurai: Shogun 2 is not the greatest, but it's certainly the most polished Total War game to date. They keep aspects of Empire and try to make it work and rightfully so. Instead of the slow town growth to grow new villages to convert to x building in your provinces like Empire has. Shogun 2 has you upgrade your castles (and only castles) to have more building slots in cities. Some provinces suck because of the introduced fertility that affects farming incomes. Some provinces are god, some provinces have special buildings, some don't. AI is actually pretty alright in diplomacy, except for of course when they should admit defeat. The battle AI is pretty meh, even on Legendary difficulty but they try to cover it up again with insane bonuses.
If you like the Sengoku Jidai era, then this is good for you. Or if you ever heard of the Genpei War which Rise of the Samurai takes place. Which sadly isn't included in the bundle, it's a nice chip in.
Fall of the Samurai: Set during the Meiji Restoration of Japan, the rapid industrialization to compete with the current world powers. Fall of the Samurai introduces some very neat mechanics. A change to the tech, where your current industrialization advances you further, but decreases public order. When you want to use samurai units compared to the rifle infantry/cannons/gatling guns/etc. SHIPS actually being useful for once in land battles, letting you bombard if they are in support range of an army. The beautiful naval combat that blows my mind away still. It's like Shogun 2 on some advanced drugs.
Rome 2 Emperor Edition: Oh boy... Where do I begin with this one. It had by far the worst start ever, even worse then Empire which is saying something. It took them 15+ patches later to fix up all the nonsense where AI starves their own armies, can't handle their economy well, battle AI is dumb as a rock, you name it. But I will talk about the Emperor Edition which is CA's fancy way of saying "Hey we fixed the game guys."
From a gameplay perspective, this is by far the biggest leap for a Total War game. Having adopted a new game engine, changing how growth works, etc, etc. You don't need a boat to transport your army anymore, as your troops will automatically turn into transport ships if you tell them to move on sea. There is no longer the super annoying and time wasting 1 stack armies from the AI just going wherever they please. As you need a general/admiral to have any troops on the field. You can recruit directly from the general as long as they are in your province. So no more moving troops over to general, rinse and repeat to build your army.
The politics is annoying and meh. It's essentially a "If you get general to x rank you can promote him to x position" all to give them modifiers. The combat is alright, unfortunately it takes far too long. As troops are no longer 1 HP, but instead have like 3-4 HP per unit and given the fact they introduced Super Heavy Infantry. Battles can take quite a while depending on circumstances. Attacking capital cities is gorgeous, there's a completely new battle map for cities like Rome/Carthage/etc. Playing this game on Legendary difficulty is just a chore to be honest. So I only recommend it if you really just want to challenge yourself against unfair odds.
Public order isn't a city to city thing, it's actually a province thing now. Where provinces usually have 3-4 cities each. The income is directly correlated to your building structures, so no more low taxes to increase town prosperity. The AI is still pretty stupid, but it's decent compared to the worst it can get. AI no longer starves themselves for lols at least. And the best part of Rome 2. The diplomacy actually means something... Somewhat... The AI is far more reasonable compared to previous titles in how they conduct diplomacy so there's a big change there.