It's all the city! Where are you trying to go and what are you planning to do?
If you've never been to LA, don't base things by the mileage distance. Traffic determines everything. Pasadena is very lovely if you want to be on more of the eastern part of Los Angeles. Old Town Pasadena area is a very nice location. So many lovely places to visit in Pasadena.
Culver City is the westside but you would want to stay right in downtown Culver for convenience to walking to restaurants etc, and it's easier access to places like Venice, Santa Monica, Beverly Hills, Hollywood. The Culver Hotel is nice. (Culver City used to be a blue collar town in the "olden days", and where movie studios were including DesiLu.) Now it's Sony Pictures, Amazon Studios, NFL Studios and a nice downtown and definitely a nice place to stay IF you stay in the walkable part of downtown Culver. *Easier access means geographically closer, not easy to get anywhere on the westside, the 405, traffic, etc.
I think of Silver Lake as trendy, funky, indie. . I think of it as a working neighborhood. Can't speak for whether it's a safe neighborhood.
NO NO NO to Van Nuys, and I'm not sure what would make someone stay in Burbank unless they really want to sit in even more traffic trying to get to touristy places.
Don't try to save money on the hotel as your first priority, if you don't have to. Get a nice, safe and convenient (as possible) location.
I love going to The Ivy Restaurant (on Robertson) for brunch on Sunday mornings if you end up staying somewhere on the westside. A fun mother/daughter experience. And there's no reason that you need to stay right in Downtown LA, because that is not particularly desirable.
If you are flying into LAX, then Culver or someplace westside is easier to get to, and if you are flying into Burbank, that's better access to Pasadena. (Again, "better" "easier" all subject to LA's definition!)