One of my favourite ways to think about the difference between WordPress.com and self-hosted WordPress is to compare building a website to building with lego bricks.
If you choose WordPress.com it’s like being given a basic bucket of lego bricks and a solid table, in a secure room, to build on.
You can build for free but your table will have advertising posters displayed on its side and you have no control over which adverts are pasted there. You’ll be able to invite friends to come round and admire your model, but the address on the invitation (your domain name) will include the word “WordPress”. There will be a range of different WordPress themes you can use (I’m not sure how that fits the lego analogy) but you can’t install your own.
Buying the relatively cheap Personal plan allows you to get rid of the adverts and the WordPress name in your address (although there will still be a discrete WordPress logo which I believe applies to all the paid plans).