Two very very different cameras.
Comparing A6000/50/1.8 vs. RX100iii-24-70/1.8-2.8.
The A6000 is very compact, with the native 50.1.8, it will fit in a small camera bag. Maybe even a very large pocket. The RX100iii will fit in a mid-sized pocket. The RX100iii will be half the weight.
Size comparison:
http://j.mp/1PjG2db
They are both much smaller than your old dSLR.
The A6000 has faster autofocus, with hybrid phase detection. The RX100ii has pretty fast contrast detect AF. For shooting action, the A6000 will be noticeably faster and more accurate. For portraits, landscapes, etc.. you won't see a difference.
The A6000 has a bigger and better viewfinder. The RX100iii does have an EVF, which is better than LCD only. But I found it tiny and awkward to use.
When using good lenses, the A6000 has better image quality potential, thanks to its larger sensor. Though I'm betting the RX100 will match or surpass the image quality of your old Canon.
But looking at the lens bundles you are talking about: The 50/1.8 on the A6000 will effectively be 75mm.
So the RX100 would give you a field of view from 24mm - 70mm. Essentially, the "normal" range.
The A6000 would give you 75mm ONLY... unless you add other lenses.
The 50/1.8 is "ok"... not great for a prime. But better than most zooms.
http://www.photozone.de/sony_nex/720-sony50f18nex?start=1
As you can see, the center is very sharp, even at 1.8. But the edges are poor until stopped all the way down to F4. Not a horrible result for a consumer prime, but not great either.
At 70mm, the RX100 will be a 2.8 lens. Considering the aperture + lens quality + sensor image quality, when shooting at 70-75mm, the A6000 will give you superior results. (You are using a half-decent lens, larger sensor, and shooting potentially at faster aperture = lower ISO).
Of course, that leaves out the whole range of anything less than 75mm. You will often find you are "too zoomed in." You will need to back up physically, you will have difficulty framing shots.
So the RX100iii will be much more versatile than the A6000.
On the other hand, you can add/buy more lenses for the A6000. The downside, the lenses aren't especially cheap... And the kit lenses aren't especially good. I dare say the 16-50, for $350, is overall a slow, poor image quality lens. So the RX100iii will bear the A6000+16-50. The E-mount 18-55 is $300, a bigger bulkier lens, not as wide, and similar image quality. To get to a decent lens, you need to go up to the Zeiss 16-70/4. That's still going to be slower than the RX100 -- and even that lens, for $1,000, has gotten so-so reviews.
So if I was purely looking at A6000 + 50/1.8 or RX100iii, I'd pick the RX100iii.
If I was willing to add lenses... but was only going to shoot in the 24-75 effective range, I'd pick the RX100iii.
If I wanted the most compact possible, I'd go with the RX100iii.
If I was willing to add more lenses down the road, and wanted greater versatility to shoot telephoto AND ultrawide, I'd go with the A6000. (But I'd be prepared to spend $2000+ over the long term).
If I wanted to shoot action, I'd go with the A6000.
