I tend to go back and forth with these games. I'll play one for a while, then the other. As I play one I see the relative flaws and remember why I was previously playing the other. It's a perpetual motion machine based on greener grass. What is it that makes me ping pong like this anyway? I mean, what's so different and so much so that I'd actually care.
Something that is different about which I don't care is workers. In IV they can cross rivers without movement cost, making them more flexible. In IV they use food, the same way settlers do, so it is tempting to delay them until city size is maxed (ignore this temptation; improved land is what helps your city grow).
Onward to the big things that make me prefer one game over the other, at least temporarily.
Vassals are better than puppets
Puppet cities don't make much sense. They're clearly part of your empire, giving you territorial control, gold, culture, and science. They have a happiness cost. They can be conquered by anyone at war with you without your enemy needing to be at war with another entity. These aren't puppets so much as poorly-managed normal cities.
While some measures were taken to reduce the habit, it's still easy to end up with an empire that is 90% puppet cities. These cities never revolt, under any circumstances. This looks absurd.
Contrast that with vassals, specifically those won through capitulation. A civilization recognizes that it is going to be wiped out, so it cedes some sovereignty to you in exchange for avoiding annihilation. You do not occupy their cities except those which you have actually conquered and kept. They are the closer thing to a puppet, being free to direct their internal affairs but following the foreign policy of their master. Under certain circumstances they can break free, sometimes with the result of war with their former masters. Puppets magically remain loyal to the very end and a civilization can even find itself with only puppets, clearly a ridiculous state of affairs.
For the player vassals also offer some advantages in terms of management. A vassal will defend itself with its own army. That makes them useful as buffers and distractions for the enemy. They will build their own infrastructure. A puppet city is entirely dependent on your army and workers. This means more distractions and more areas needing management. All they save relative to a annexed city is the production orders, but that comes at the cost of being unable to manage population; I'll trade some more micromanagement for the ability to avoid the AI-run city growing out of control. Though, I could use my workers to remove all food-producing improvements around the puppet, so then I've not saved any micromanagement after all.
Finally, I don't like having to keep playing after I've clearly won. Capitulation means that you can deal with a civilization and be done with it without needing to slog through every single city to make an empire of puppets. When you're the dominant power civilizations will tend to capitulate faster, so that again you don't need to slog through every single city to win. You can return captured cities to their civilizations rather than waste effort and resources to develop them to be useful to you.
On this one I have to give the clear advantage to IV.
Tactical combat is fun, but clearly cheating
I prefer the tactical combat of V. It's more interesting to me than smashing stacks into each other. However, it also feels like cheating. The AI is not very good at it. I can't say that the AI in IV is brilliant, but the stack nature means that there are limits to how much I can run circles around it. Economics are much more important in IV as well, since I am going to lose units; even with air superiority there is only so much I can do before I have to risk actual units. Nuclear weapons can eliminate causalities (on my side by making them 100% on the other), but they are automatically killed, so it's not entirely accurate to say that there are no losses.
Any time I've found myself facing multiple economically-superior opponents in IV I find myself longing for the tactical combat of V. It's a handy crutch for bad economic policy, or bad luck. Yet is that crutch a good thing?
I can't quite point to one being better than the other with combat. Combat in V is more fun, but in IV it is more balanced for the AI and makes economic decisions more important.
City States and Trade Routes
I like city states. They give us something to do to hurt other civilizations that aren't war. However, the AI isn't very aggressive about keeping them on their side, so these tend to screw up the diplomatic victory, which is terrible in both games, and the congress in general. The AI used to try harder to keep them but that changed at some point, I suspect because people got mad at perpetual coups and the AI's bottomless treasuries. City States also give us friends of last resort, meaning trading partners of last resort, and can act a bit like the vassals of IV in terms of acting as buffers and distractions.
Trade routes, though fairly recent, make it feel like a more complete game. They add more layers to diplomacy: losing a trading partner is to be avoided, as is having none at all. They offer some interesting choices: maximize gold, science, or spread of religion, or use them internally to boost production and growth.
Obviously this goes to V.
Gold
V gives the player something to do with it. Bribe city states, buy units, buy buildings, and so on. It is a currency that changes how you play depending on how much you have. However, buildings carry a non-trivial maintenance cost, making infrastructure a risky investment. IV instead allows you to trade gold for science or culture production, giving greater flexibility. You can sell off the buildings in V, but that is a ridiculous level of micromanagement for something that used to be a straightforward tradeoff.
This one is going to V. While the science-gold trade is a less-annoying mechanic, the general uselessness of gold in IV has bugged me. If players could generate a lot more with the rush function from universal suffrage, then gold would be worthwhile.
Interface and graphics
I group these together because the usefulness of information depends on how it is displayed, which is in turn affected by how it can be displayed, meaning the graphical limitations.
The outdated maps (meaning that they don't update until you buy a map or send a unit to look) of IV tend to bug me. Expansion is hard to manage when it is not clear where the borders are. Keeping scouts out there is expensive with the supply cost and a general annoyance. It's harder to see civilization borders in general on the map, perhaps because of the graphical limits. On the other hand, V has a bad habit of not diplaying units or updating tiles to reflect roads and rail, so I'm frequently using f10 to get the hex display, which is ugly and a stupid thing to have to do.
On the other hand, IV has far superior production and happiness management. Being able to queue up multiple buildings is a great thing. Being able to queue up an endless run of a unit makes it easier to manage the logistical side of war. Not having three tiers of what are essentially just barracks means fewer hassles when gearing up for war. Of course needing to switch civics might sound like a hassle as well, but it makes more sense to have political changes to support war than to have, for thousands of years, been on a cultural war-footing.
The combat log in IV is pretty useful for figuring out what happened after dozens of battles scattered across the world. Even better, clicking on a note in it will center the screen on that location.
IV puts AI conversations at the start of the turn rather than the end. This means that if you misclick you can load the autosave and none of your decisions are lost. V places the AI conversations at the end, so you're stuck with either redoing an entire turn or saving the game at the end of every turn. Neither of those are good.
IV tends to look universally ugly regardless of the circumstances, partly due to the graphics quality and partly due to the mess of roads and rail. V is prettier and more nuanced. It shines and sparkles, but cities also burns, the pillaged landscapes are charred and smoking, and nuclear fallout adds a haunting glow. Fallout in IV just looks like any other ugly tile with a patch of ugly layered on top.
Perhaps I should not have combined these. Interface goes to IV, graphics to V.
Performance
IV loads a game in seconds, V can take several minutes. Given how much a misclick can screw things up in either game, but
particularly in the tactical combat of V, loading a turn is a huge deal and not an uncommon occurrence. Turns in IV can take very little time, particularly if there are no enemy movements to watch, while they can drag on in V. The start times are much different between the two as well. The only performance issue that I see in IV is when calculating damage from nukes, and since those tend to come out near the end of the game, if ever, this is rarely an issue.
This one obviously goes to IV.
Conclusion
All things considered, there are parts that I like in both of them. Since it is the newer title, V is going to get more developer attention and will most likely continue to gain relative to IV; in its initial state IV was clearly the superior game. If V ran better, and perhaps with a newer computer I'd see significant gains, then I think it would become the superior game by a non-trivial margin. Until then, I think it is entirely a matter of personal preference.