There's a lot of differences, I don't expect I'll remember them all off the top of my head, but I will try to summarize for you. Keep in mind that HoI3 today is after 3 major expansion packs, each with a year of dev time, so you're comparing the final form of HoI3 to the in-progress Hoi4. A lot of major mechanics were only added or fleshed out to HoI3 with these expansions.
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In Hearts of Iron 3, there was no "World Tension" meter, instead, being able to declare wars and pass laws such as conscription or mobilization was based around "threat" and "neutrality". Each country had "threat" with each other country. If Germany created a lot of threat, and the countries around Germany had low enough "neutrality", they were able to declare war.
In Hearts of Iron 3, there was a resource called "leadership" which the player needed to channel into Research, Espionage Points, Officers, and Diplomacy Points. You controlled how much leadership went into each thing. 1 point of leadership could research 1 technology, or generate a certain number of officers per day, or generate a certain number of espionage or diplomacy points per day. Small countries would have 6-10 leadership, large countries like Germany would have closer to 30.
In Hearts of Iron 3, there were a huge number of technologies compared to HoI4 since you could research many technologies at a time. Most technologies were "repeating", with years attached. For example, "infantry small arms" would have a 1936, 1939, 1941, 1943, 1945, 1947, 1949... version.
In Hearts of Iron 3, there was no separation between Military, Civilian and Naval factories. There was a generic resource called "IC" which built everything, and there was a single production queue for IC, ships, aircraft, land units and so on. IC was assigned to - "Production", "Supply", "reinforcements", "upgrades", "consumer goods".
In Hearts of Iron 3, there was no such thing as "equipment", land units simply had a certain number of stats based on technology. If you build a unit with 1941 equipment, and wanted to upgrade it to 1943 equipment, you needed to assign IC to "unit upgrades" and they would slowly be upgraded. Similarly, if a unit was damaged, you simply assigned IC to "reinforcements" and the damage would be repaired as long as you had enough manpower.
In HoI3, there was no "production efficiency" in the same way there was in HoI4, since there were no production lines. Instead, you had a system called "practicals". Building tanks increased your Armored Vehicle practical, building infantry increased Infantry Practical. This boosted research speed for related technologies, and reduced the build time for future units of that. Practicals faded over time without use.
In HoI3, manpower was a steady income instead of a flat percentage of population. With a high conscription law, your manpower would increase at some abstract number like "2 per month", which might represent 2000 or 20000 men or something like that. In HoI4, you pass a conscription law and all eligible people are granted to you in manpower, but the increase over time is very slow to represent population growth.
In HoI3, you had a full Order of Battle. Every division could be assigned to a corps, which was assigned to an army, then an army group, then a theatre. Each division, corps, army, army group and theatre would have its own leaders.
In HoI3, there was a simple espionage system. You assigned spies to enemy countries, and they would give you intelligence, or fight enemy spies, or do coups. In HoI4 currently there is no espionage, however parts of the old system are incorporated through "decryption/encryption", and there is a system for coups and influence separate from espionage.
In HoI3, air wings were manually controlled and told to cover certain areas or fly to specific provinces to do bombings in support of specific attacks. In Hoi4, you assign to air-regions and tell them to perform certain missions. In Hoi3 air wings had HP, while in Hoi4 they are modelled as individual planes that are lost and need to be replaced.
In HoI3, there was a full, detailed (but also buggy) supply system that traced supplies that you built from your capital or a local supply depot to the front lines. In HoI4, supply is a bit more abstract, you don't build them directly, they come from regions and flow. It's simpler and less impactful, but less buggy and doesn't require as much attention.
In HoI3, there was no concept of unit training or early deployment. You simply had laws that controlled how highly trained units were. You couldn't emergency deploy, nor could you perform on-map training on units to boost up their XP.
In HoI3, you did not have equipment or equipment stockpiles, but you did have resource stockpiles. Metal, Energy, Fuel, Crude Oil, Rares, Supplies and Money were the resources you had, and you traded for them. Any surplus was stockpiled allowing you to build up massive stockpiles pre-war that could last you for years. In HoI4, resources are not stockpiled, they are simply in surpluss or deficit, and control how fast you build things. In Hoi3, you needed 1 energy, 1 metal and 1 rare for every unit of IC.
In Hoi3, units used both supply and fuel. In Hoi4, fuel is abstracted into production and units only use supply to move.
In HoI3, there was only "AI control" or full manual control for units in combat. HoI3 was mainly designed for full player control with non-critical fronts being relegated to the AI. In HoI4, the battle planner is AI controlled but designed for constant player intervention alongside the AI. In HoI3, you could set objectives for the AI but it was quite poor. In HOI4, you set frontlines and tell the AI where to advance. It's an improvement over hoi3 ai control, but still has problems that need fixing.
In HoI3, there were no peace conferences, just "war goals" that you added during the war. If you ended the war and forgot to set war goals, the war simply ended in white peace.
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Overall, in my opinion HoI3 is a more complex game, but not necessarily better. HoI3 version 1.0 at launch was pretty bad and a complete technical mess, HoI4 1.0 also had problems but was much more stable and well rounded. Certain aspects of HoI3 (with Their Finest Hour expansion) are superior to HoI4 as of today. I believe that HoI4 will eventually surpass HoI3 in almost every area, and that currently it is a more fun game to play. But this is just my perspective - HoI3 is also a more challenging game, has a higher degree of complexity (for better and worse), so some of the more hardcore players will consider HoI3 to be more fun.
