I'll focus on what I think are some of the biggest flaws, because that's easier to talk about:
Social policy trees are completely unbalanced. Tradition is the best pick in the vast majority of circumstances. Piety is awful, honor is pretty poor, and liberty is okay but still nearly always outclassed by Tradition. Rationalism should nearly always be picked as soon as you have it available. A proper rebalancing of social policies, buffing the weaker ones, would make the game SO much better.
They've tried rebalancing over the years, and while there have been improvements (Tradition used to be much weaker relative to Liberty than Liberty now is relative to Tradition, and Piety is a lot more useful than it was at launch - although arguably it was at its best in G&K), there's a core design issue with the specific trees they settled on.
Tall vs. wide is a bad decision to force for an early-game tree because you're often not going to have enough information on how widely you can settle to know whether Liberty is a good choice; it was in the early stages of Civ V only because it gave you more benefit even for a small empire (by rushing an early GS and getting a fast second city) than Tradition did at the time.
Resource-specific trees are also problematic - especially for food and science, the two most important resources, but on the flip side also for faith because the value of that resource varies very much depending on both your strategy and your ability to get an early religion. Piety is much stronger on lower difficulty levels - the problem isn't with the tree so much as with the strategies required at higher difficulties, in which Piety comes too late to help with pantheons or getting an early religion and doesn't reward you enough if you have a late one.
The game punishes expansion far too much, and the majority of terrain is so crap you wouldn't want to settle there anyway.
For all of the game mechanic improvements with BNW, in some ways the game was at its best in G&K - wide vs. tall was as balanced as it has ever been in a Civ game, before BNW over-penalised expansion (with, ironically, a better mechanic for constraining expansion than the game's previous system), and both warfare and peaceful play were fully possible (with the series' typical preference for the military approach). Though I wouldn't say that war is punished so much as not sufficiently rewarded - puppet cities aren't especially useful, there aren't resources worth fighting over, and Natural Wonders are almost all at their best early in the game, and by the time you've built an army and captured one the major benefits you'd have got from it are lost.
In almost all my games I never settle a fifth city, and in a lot of them I don't even settle a fourth because there's no good terrain that's worth the settle left. War also rarely feels worth it. Sprawling empires in Civ 5 just aren't that much of a thing.
Religion may as well simply not exist on higher difficulty levels. An entire aspect of the game that's great to play around with in lower levels is just something that, for the most part, just passively happens to you on higher levels.
I think religion needs some way for multiple religions to coexist (i.e. benefits from multiple religions in a city, in Civ V only possible with one Piety policy and the Indonesian candi).
There should also be some ability to alter the properties of a religion you inherit (so that you gain a benefit relevant to your strategy in place of a benefit the founder receives, for instance) - this would also allow possibilities for religious tensions to emerge diplomatically. Reformation, for instance, could be an option open to any civ following a religion rather than just the founder (the current system is akin to saying that only Rome can reform Christianity, when in reality the Protestant Reformation happened in Germany).
Cities lack distinction or specialisation. They all feel roughly the same. There's a lack of depth in individual city management.
There does need to be more to city management (local public order, taxation, perhaps immigration/emigration along trade routes), but specialisation is certainly there - in most games you're likely to have trade cities and will probably eschew market-chain improvements in any city that doesn't produce useful amounts of gold; you're likely to concentrate your cultural national wonders in a single city; and you're usually going to need a particular city as a main production centre. This covers all the bases previous Civ games employed, without going the paint-by-numbers route of Rome 2 or Civ IV (i.e. stack all the modifiers of the chosen colour in the same place).
Science is too important compared to other yields. In order to win a culture or domination victory, you essentially have to win a science one as well (except you just don't bother with the spaceship).
This is raised as a criticism again and again, yet it's true of every single game in the Civ series (save, from what I've read here, Civ Rev) - and that includes the original Avalon Hill board game it's ultimately based on (in which science was the only victory). I'm not sure why so many people are hell-bent on Civ V 'fixing' this, if it's even a problem for a game that's fundamentally driven by progress up the tech tree. To some degree the game is less dependent on science than the previous iterations, since there's no direct relation between the rate of technological development and unlocking civics/governments or religion (obviously the higher-tech your production structures the more of each resource you accumulate, but then that was also true in the older games in addition to the policies and religions themselves being unlocked by technology).
The only difference between Civ V and the older games is that the combat system in Civ V makes it even harder for a lower-tech civ to beat a higher-tech one in warfare than it was in its predecessors.
Unbalanced tile improvements. There's not much choice involved on what to improve, usually. Farms are almost always the best option except on hills. Mines are usually the best option on hills. Really the only time I think about tile improvements is a hill adjacent to fresh water. Trading posts are just useless (except in forest).
I don't particularly agree with this - farms are usually better than trading posts because of the extra value of food relative to gold, but posts are important for cities with market chain buildings since these no longer give meaningful flat gold yields, and later in the game add useful science. And in forests they do compete with lumber mills, which is a meaningful decision.
I miss the number of improvements to choose between in Civ IV, but for the most part all those gave you were larger numbers of non-cottage options to ignore.
The main change I would make to the Civ V system is to reintroduce the original Civ irrigation mechanic, which would promote both settling along rivers more frequently (there can be a food bonus, but unless you have a river through grassland this isn't producing extra food over a typical grassland location, and the trade route bonus is only relevant for inland trade cities) and diversifying improvements. Of course, you then need to add another improvement type so that there are options other than cottages - making shrines a tile improvement might work, and/or culture improvements (already available for a couple of civs).
Bonus resources are boring and just feel like a slightly better normal tile. They should have an actual function like luxuries and strategic resources do.
The AI are far too predictable in the World Congress. Everyone likes arts focus, everyone hates science focus. Everyone likes the collaborative projects. Nobody ever votes for anyone other than themselves.
The first two are irritations of a clumsy diplomatic AI - the latter is crippling to gameplay, and worse is specifically hard-coded so that the AI can't actually vote for anyone else for, say, world leader unless bribed (and then will contribute the minimum number of votes). What's more diplomatic relations aren't considered in the calculation at all - a civ that has no preference either way on an issue won't vote for an ally to keep good relations, for instance, nor will a civ ever favour maintaining relations over voting for a measure they want even if that measure is unlikely to pass ("Yes, I knew science funding wouldn't pass and everyone would hate me for it, but since in principle it would benefit me if it did pass, I voted for it").
It's too easy to get free diplo points by choosing policies you like or passively voting for someone else's policy that's guaranteed to pass anyway.
That's fine - it's the way the real world works, and the way Civ IV did it as well (though better would be if there's only a significant diplo bonus if the measure wouldn't have passed without your votes). The problem is that it's very easy to 'game' when the AI doesn't do the same thing.
- and preparing for war.
