ELI5: The difference between POE.trade and an Auction House

The subject of trade and trade tools gets brought up a lot and there seems to be three general views:

• People who want an in game auction house
• People who want the functionality of the API to be extended for sites like POE.trade
• People who want as little trade as possible

I've seen people say they don't want an Auction House in game but in the same post say they would like to see the functionality of POE.trade extended. I don't really understand the difference.

I'm a fairly casual player so it's possible there are some fundamental things I don't understand but I use POE.trade exactly like I use an Auction House. I look for the item I want and try and buy the cheapest version that fits my requirements. Sometimes its by searching for item names, sometimes by specific affixes/suffixes.

I don't get why poe.trade is fine but an in game auction house isn't. Is it just the idea of having it hidden away from general view puts a barrier to entry on it so it stems the flow of items? Is it because it is easier for GGG to iterate on an API than it is to develop an in game system? Is it something in the functionality of either POE.trade or an in game AH?