Personally, I found 4 to be slightly superior to 5
4 Pros:
Much better (IMO) economy, micromanaging, tech tree, and expansion
The fact that each city has a individual health/sickness bar made it so much easier to build/plan, and it allowed you to really plan out your cities in a helpful way.
The espionage/diplomacy was a lot clearer and consistent in 4, making it easier to form trades/alliances in it.
IMO, the religion system made more sense in 4, with the world being divided into religious factions that tended to aid their brothers/sisters in faith and defend against any others.
Cons:
The military system in 5 is far better, whereas 4 had the 'stack of doom' problem.
5
Pros:
The military system in this is far, far better-although there's some realism problems with ranges, the fact that cities can defend themselves and the one-unit per tile thing actually gives the positioning of troops meaning. Honestly, in a game as abstract as civ, the realism doesn't really matter to me.
Being able to rush-buy any (non-wonder) thing makes a massive difference, giving a real reason to save up money. Honestly, it's the highest praise when you ask 'why wasn't this in the series all along?'-it fits in perfectly.
Graphics are (obviously) better
The map-generating system seems better in 5-creating mountain ranges which divide continents and civs and so forth. The maps on 4 just feel poorer now.
City-States, I guess? They've never really been important to me, and when I go for a diplo win it's always just a case of buying them out, so take that as you will. They don't really detract from the game at all, so I guess you could put them in the 'pros' category.
Cons:
The diplomacy can be really, really arbitrary sometimes. Once, I was asked to attack a civ- Austria, IIRC. While I was moving/readying my troops, Austria messaged me to say they'd noticed the movements, and asked if I was about to declare war. I responded that I was, only to have the civ that asked be to attack Austria in the first place denounce me for 'warmongering', which derailed my 'diplomatic/religious world government game' into 'wipe literally every civilisation out with my army, which no-one can even touch due to my ridiculous economy'.
I personally found 5 to be easier-on 4, I would stay round noble, maybe moving up to prince occasionally. On 5, it's king/emperor before I'm getting the same win/lose ratio.
I really dislike 'global happiness' and the luxury resources system, and found that all it did was make expansion harder for no rational reason. Why does having more cities make people unhappier? Abstraction is one thing, but the lack of any rational link is another.
Main differences:
In 5, the military system is more in depth and detailed. This can, however, lead to situations where the player has a massive advantage due to being human, and can wipe the floor with AI armies, so take that as you will.
The biggest 'not good or bad, just different' difference is in the religions-in 5 (with G&K), you can make and shape your own religion, giving you more options. However, I was personally a fan of the religious system in 4-it was a lot less detailed, but it shaped how you saw the world to a massive degree, as you would almost need to be the same religion as an AI you were trying to butter up, and the religious factions that would form made the world feel more 'alive'.
I've spoken before about the diplomacy system, but let me re-iterate just how much I loath loath loath the diplomacy system in 5. It feels like you're blundering about in the dark, with no real feedback for actions. Declarations of friendship, co-operation, denouncing etc just makes it confusing. I need a declaration of friendship to trade gold? Why? Fine, if they can't offer me what I want, I guess I'll just let the Aztecs overrun the bastards. It's (slightly) better now that you have the ability to see what they like/dislike, but the numbers game in 4 just made more sense. Yes, it was 'gamey', but as I said, I don't really care too much about realism.
The diplomacy in 5 can sometimes ruin the military side of the game-I once had the entire world DOW me for 'warmongering' because I wiped out England after they attacked me , then 10 turns later each individually peace out for all of their non-capital cities. Then I wiped the floor with them one by one.
The warmongering system in general in 5-IMO, it was a valiant effort, but didn't quite work. It doesn't distinguish enough between defensive and aggressive wars, and unduly penalises wiping out civilisations/taking capital cities, even if they repeatedly attack you. Sure, I could leave Monty with his one tundra city, but then he'll just try to attack me again. For the 7th time. Because I own his capital. Which I took the 4th time he attacked me. And no, I will not give it back. However, if that doesn't concern you-you're a science/culture player, for example-then you might like it. I do like the idea of diplomacy penalties for warmongers, though.
The economy in 4 was based on cottages and/or specialists, the one in 5 is based around trade routes. Honestly, this doesn't make too much difference to me. It does make isolated starts easier in 4/harder in 5, though.
In 4, the AI will throw up far more cities, but they mean far less-in 5, I tend to get around 4-5 cities/civ, in 4, it could be in the mid-20's. I personally like having big maps with lots of cities, but you may not. Also, if you're regularly conquering cities in 5, you'll have to raze them to avoid happiness caps from killing you, whereas in 4 you could keep a 'frontier' of miserable, angry, sick cities between you and the enemy.
Lack of technology trading. I liked being able to trade techs, but they got rid of it. I guess some people might like research agreements better? Subjective, of course, but I liked the tech trading, even if you did have the 'tech to alphabet, immediately backfill all the tech's you missed, become miles ahead of the rest' problem.