Just like London it is a collection of "villages". What were originally separate cities: Santa Monica, Hollywood, Pasadena etc have all been absorbed into the whole that is LA. I'm sure you can recognize the same process in london where Hampstead and Stratford have distinct personalities and yet are now both a small part of the whole that is London.
Now LA, unlike London is a city much easier to live in and less easy to experience as a tourist. Tourists try to travel from Malibu to Hollywood boulevard to the Broad museum all in the same day. This is a wonderful way to experience LA traffic. Any person who lives here would recommend you choose on area and spend most of your time experiencing it properly. If you want a walkable neighborhood go to Santa Monica, West Hollywood or Downtown. If you want nature, hiking and to feel like you are not in the city go to Malibu or the Hollywood Hills.
I would argue that all of the LA neighborhoods I mentioned are very familiar to most of us because we watch movies and television and LA is traditionally where those products are made. Maybe you don't watch tv or go to the movies - in that case go to the David Hockney exhibition in london and you will see that LA is one of the most distinctive cities in the world. It is good weather, easy living, good looking inhabitants and it is surrounded by a natural beauty few other cities possess. I could argue it makes London look like a giant rat hole in comparison but to each their own!
I think London has its historic sights and some pretty neighborhoods but the livability of the city is shockingly bad. From crappy housing stock to antiquated tube - I am much happier in LA. But that's just me.