I have an older Cateye wireless bike computer and I usually ride with Strava running on my iPhone.

I find the bike computer very accurate, WHEN it registered. Being an older wireless unit, it will occasionally get interference and speed shows up as 0 or something much lower then I am traveling, which impacts distance. This is typically enough to loose a few tenths of a mile to a mile on a longer ride. I assume this would not be an issue with a wired computer or a newer ANT+ wireless computer, but who knows.

Strava seems to give me very consistent distances every day. The only catch is to make sure it gets my location locked before I start or it might mess up the first part with a sudden big move. Having wifi on and giving it 5 seconds or so usually takes care of this for me. I have tried some similar apps as well (map my ride, etc.) with good success as well.

My only complaint with Strava accuracy is elevation - its not consistent, maybe a 5-15% variation on the same route I ride regularly. no idea why. On rare occasions, its way off, 20-30%. Typically on these days it puts me on the map as though I rode 20-50 feet off the side of the road, which is often down a big drop and messes with elevation readings I guess. This isn't common though.

I guess a bike computer with an altimiter would be more accurate.

Not sure how well a dedicated GPS bike computer would be in comparison - proboably better then either of the above as it could have sensors plus GPS, and software could smooth the two out.