Has anyone ever read American Cinematographer magazine? The holy grail of cinematographers is depth in an image, usually through focal length and lighting. It's not about using a Panaflex body, or an Arri because Panavision sucks and is overpriced. They go into tremendous detail about lighting...even if it's supposed to look like natural light.
Translate this to still photography.
I think the reason I use more professional equipment for personal photography is because I think it's easier to add depth. This frequently has to do with sensor size (notice I'm not talking about image noise...I really don't care about that...I've used fast film before) but mostly with the glass. My favorite vacation photos have depth, and even though I've taken wonderful images of my family with a p&s in a dark restaurant, they're wonderful because of the subject...not the merits of the photo itself.
I think you can make images technically better by lots of post-production, but it won't make up for the depth you can create with gear that allows separation and control. We're not talking snapshots, we're talking portraits or other planned photography.
So, for Nick (and by the way, your wife is a beautiful woman, so understand we're talking about your posted subject), the image seems kind of like all the stops were pulled to salvage a snapshot. Please don't take that as a personal or professional attack. I'm just talking about the technical act of capturing images with two different tiers of equipment. I happen to think that it would have looked much better (more organic) with a better lump of glass, larger sensor, less compression, and maybe (though not specific to the camera) more depth through lighting.
So, long-winded opinion...yes, the gear CAN matter.
Family snapshot with decent sensor and lens combo (it's not art, just a snap):