What part of NY did you live in? I lived there about 4 years ago and can draw a pretty good comparison between neighborhoods. I'd say that what makes Diamond Heights materially cheaper is the fact that a lot of the townhouses look the same. It doesn't quite have the character that most San Francisco neighborhoods have. Don't get me wrong, it has good value for the convenience and proximity to downtown, but I'd say that most San Francisco home shoppers look for a little different criteria.
Now to answer your question. It is on Twin Peaks and faces west. It is on the same ridge as Diamond Heights but at a higher elevation. Same weather pattern. It is also an area built with auto commuting in mind. It is a residential area with no neighborhood shopping. The closest shopping would be the Mollie Stone Tower Market. Which is a great store. The Safeway in Diamond Heights is the large supernarket.
I live in Miraloma Park and grew up in the area.
Aside from public transportation all of these areas are very central to the City and the freeways to head south. There are busses that service the area but not like other parts of the City.
Diamond Heights was built in the '60s and has many large apartment complexes that were turned into condos with pools and workout rooms. As one of the last areas inside the city to be developed it has to say something. There are also many single family homes on the hills there. The rents are less expensive in the complexes because the style of the buildings are more apartment like than other San Francisco neighborhood rentals. You have long hallways and elevators, it feels suburban and more hotel like than SF neighborhoody.
The weather in SF is very variable. It can be in the 70's downtown and 10 degrees cooler at the beach. If you have a map of the city draw a line from just east of the Golden Gate Bridge heading south across Twin Peaks and Mt. Davidson to the west is fog and the east sun in a typical summer weather pattern. The Pacific fog sits on the Ocean and moves in during the night and then burns off during the day. The area to the west of the line is sloped gently to the Ocean as the land rises it the fog is forced up into the warmer air and burns off. So the first ridge is high enough to lose the fog layer. Diamond Heights is on the top of that ridge. Glen Park and Noe Valley are below it to the east.
Safety is such a personal decision that no one can decide if you would feel safe other then you. When you look at the crime stats remember that you are looking at a compilation of every thisng that is wrong with humans. They don't document all the good or mundane activities that happen.
I moved from NYC about 20 years ago and love the City. I moved to Noe Valley, which is just below Diamond Heights. Much of Diamond Heights has great views of the City, and there's a Safeway grocery store which anchors a strip mall with a post office, coffee shop, chinese restaurant and more. It's located between Glen Park, Miraloma Park and Noe Valley -- so you have proximity to the small-town downtown areas of Noe (24th Street) and Glen Park (Chenery) as well as the upscale Molly Stone grocery in Miraloma Park.
Sally's suggestion of the police maps is a good one to follow up on. I think the rents are a bit less in Diamond Heights because it's further out from town, and although there are some bus lines it's not as connected to public transit as some other neighborhoods, and it is foggy compared to Noe Valley and the Mission. Welcome to San Francisco!
Foggy weather can be a big issue there. For safety issues you need to go to the police website which you can get to via the Mayor's website - http://www.sfgov.org. My website also has a direct link to the police via the Links page.