You are correct that they both work very similarly. In fact doge was heavily inspired by BTC. Both have PoW and coin minting as a result of block resolutions.
Granted I'm not an expert on doge, but one big difference that I know of is the limit of coins that will be minted – bitcoin caps at 21 million, dogecoin is uncapped.
That's a rather important fundamental when talking about a currency.
I would then point to the communities that hover around either BTC or Doge. From personal observation, BTC is more serious about digital currency as a possible store of value future, whereas doge is a joke / meme focused coin.
At the end of the day, a lot of it is marketing, which is why I actually thought Doge would be a possible contender for a 3rd place crypto coin around the time when Elon Musk was tweeting about it. I believe the fact that it is uncapped and that is really is more of a meme coin than anything serious (see the community around it) led to its downfall. It didn't have solid enough legs.
Perceived value. It worths what people are willing to pay for. A lot of things work that way. Skeptics might wonder how long it would last, but less about how much it should be worth. People aren't only buying tech.
We do have a lot of scammers and useless coins, but tech is only one indicator of value, not the ultimate method to predict perceived value.
