Hi Mogwai. Am not stalking you honest.
I've lived in Bondi Junction and am currently in Coogee.
It depends what you are after really. Coogee is very close to Randwick, so great for the hospital (walking distance). Coogee is a beach town, lots of restaurants and cafes and of course, the beautiful beach. It does get backpackers but I love it here.
Bondi Junction is about 2 miles inland from Bondi Beach. A brilliant (quite posh) shopping town at the end of the line for the underground, so superb links into the city (Coogee doesn't have this, which is why it's a bit cheaper). It's on top of Woollahra, which is very posh, and Paddington, also posh. All of these are 10-15 minutes away from Sydney central. Great locations.
Bondi Beach is quite young, backpacky but has a good vibe. I enjoyed being close to there. The surf is too strong for toddlers but there are some good rock pools (paddling pools) there that are suitable.
I'm happier in Coogee than I was in Bondi Junction, Bondi or Woollahra, but mainly because I'm right on top of the beach and have everything I need within a 5 minute walk.
I go to Bondi Junction to shop, it's about a 15 minute drive away and has everything you would ever need - biggest Westfield shopping centre in Australia I believe - very smart and modern with a cinema at the top. I saw Harrison Ford and Calista there a few weeks ago.
Kensington has amazing houses - huge, huge great places with swimming pools, cellars and bars. It's quite isolated for public transport & walking (good golf course though), but great for a car. If you ever fancied buying a million pound house in the UK, you'd find one like it here.