There's a few things to note: (I'm assuming you're playing the complete edition with all the expansions)
Combat:
1. Cities can't defend themselves. If it's empty any military unit can capture it without a fight.
2. Units can stack, and stacking is important. In Civ 4 a large stack is usually called a "Stack of Doom" and that's not exaggerating in any way. I have a more in-depth explanation below.
3. When combat starts, units fight until one dies. They don't disengage after a round of combat like in Civ 5. Some units have a retreat ability though, which may cause them to disengage when they're basically dead, but it's usually only found only on mounted units and isn't very common.
4. There are no ranged units, except aircraft.
5. Units have bonuses in specific circumstances. Archers get a city defense bonus and a hills defense bonus. Axemen get a +50% strength bonus when fighting melee units, spearmen get a +100% strength bonus when fighting mounted units, etc.
6. When getting a promotion units are healed for free on top of getting a bonus.
7. Promotions are better in Civ 4 than they are in Civ 5, and promotions tend to unlock even more broken promotions. So defend you experienced soldiers, they can be worth a lot.
Economy:
1. Everything is non-centralized. Every city has it's own happiness rating for example, it's not spread out thorough out your empire.
2. Roads don't cost upkeep, so build them on every tile.
3. There is a healthiness/disease mechanic. It's much like happiness/unhappiness. Having more unhappy citizens then happy causes some of them not to work, and having more unhealthiness than healthiness causes the city to lose food.
Edit: 4. The "coins" collected in the city tiles aren't gold. They're commerce. Commerce is split into gold, science, culture and espionage based on what you set it to do. The little percentage markers at the top left of the screen lets you decide which to go for.
Edit: 5. Gold is harder to get in Civ 4, and has less uses. Gold in Civ 4 is mostly a balancing factor to make players develop their cities rather than just build more, as more cities increases your upkeep cost a lot. Especially when they're built far away from your capital. So long as you always have a couple of hundred in your bank account that's all you'll need, unless you're running a specialized empire build.
Stacking and advanced combat stuff:
Units will always fight against the unit with the best chance of survival in the stack it's targeting.
If you attack a stack you might lose your unit and only hurt one of their units, then the unit that you hurt will drop to the "bottom" of the stack, meaning that you have to kill the rest of the units before you can kill the one you damaged.
That means that you have to make sure you can destroy the entire stack you're targeting. Killing off your entire stack to only hurt all the units in an enemies stack is a complete loss.
I'll sat that again:
never attack when you can't start killing enemy units, because the units you hurt but do not kill will become very low priority and be protected by the other units in their stack. If there are two equal stacks, the first to attack is the one that dies, often without killing a single unit in the opposing stack. And the surviving stack will have gotten a lot of experience for it and be stronger than ever.
With the Beyond the Sword expansion Siege units can't kill enemy units but deal splash damage to multiple units in the target stack, which is extremely valuable. Attacking a stack of 5 units with a catapult won't kill anything, but it will soften up all of the targets, making the rest of the units in your stack have a much better change of killing things.
So if you're playing the complete edition, siege weapons are extremely useful but disposable units.
Mounted units have a flanking bonus in Beyond the Sword/Complete Edition. It allows them to deal "splash damage" to only the siege units in an enemy stack when attacking. So they're the one way to stop siege units from messing up your stack. (other than smashing their stack with your own siege units and then destroying it with other units)
Because of all that, defense is easier than offence unless there are siege weapons present.
Also, it's generally best to make very diverse stacks. Don't invest in only swordsmen, (6 strength, +10% city attack) because even if you make 25 of them your enemy might make 8 axemen (5 strength, +50% vs. melee units) and easily hold them all off. (Sounds funny to some people, but if the swordsmen attack they'll likely lose 8 before they start being able to attack even slightly wounded axemen, and will have likely lost 16 before they kill a single axemen. In the end even though they outnumber the axemen 3 to 1, they might all die and leave 1-2 axemen which will have
huge amounts of experience after the battle)
There's probably more, but it's hard to remember everything because the game has a lot more little things to keep track of and tweak then Civ 5 does, at least in my opinion.