I simplified what Al (my husband) told me. He spent 30 years as a Land Surveyor (after several as a construction surveyor) and is my go-to in such things. 

I've been retired for a year now, but in my career as a Professional Geologist and Professional Engineer I relied on GPS for many projects.  The accuracy of measurements made using professional GPS equipment and techniques that tie into land-based systems is far superior to the portable GPS measurements made by our bicycle GPS systems.

Assuming your cycle computer has been properly calibrated for the wheel size,  it will always be more accurate than a bicycle GPS measurement of distance.   GPS instruments don't work continuously, there is some finite time between each measurement (each "pinging" of the satellites). When you go through a tunnel or under dense tree coverage there are gaps in the measurements. Incidentally, what a GPS actually measures is time - your position is determined by triangulating from several satellites in different orbits based on the travel time of the signal from the satellite.  The accuracy of the position triangulation depends on the number of satellites visible to your GPS receiver at the time of measurement. The GPS software then has to make an assumption about your path from the last location to the current one - a straight line guess may or may not be appropriate. GPS measurements can be quite accurate under ideal conditions, but they will never be as accurate as a cycle computer that is continuously recording each revolution of the wheel.

All that being said, on my Northern Tier tour last year I was surprised that my riding partner's GPS and my cycle computer almost always agreed within a mile at the end of the day.

I've never run both at the same time - I switched from a bike computer to GPS (Wahoo) trip computer/mapping device.

As you and others point out, if you accurately measure the tire circumference (vs. just use tire size) the bike computer is probably going to be more accurate - if the sensor stays well positioned and doesn't miss any revolutions, etc. Also, not sure of the math in bikes but for cars a few PSI change in tire pressure can change the diameter of the tire 1/4 inch or more, which would impact distance measurement a bit. 

But, since I'm planning routes on GPS and measuring distances on GPS, don't suffer that disappointment of more miles to go!

Being something of a belt-and-suspenders type, I like having some level of redundancy.  So, although RWGPS gives me both route and distance traveled, as well as a bunch of other data for which I have little use, I also mount and use a cyclometer and (when on tour) a dedicated GPS.

Batteries quit.  Devices overheat and stop working.  Things fail, or fall off and get run over (or lost) before you can retrieve them.

I'll use the bike computer distance- which always exceeds the GPS-indicated distance- as the determinant of how far I actually rode on any given day, and for the trip as a whole.

"The accuracy of measurements made using professional GPS equipment and techniques that tie into land-based systems is far superior to the portable GPS measurements made by our bicycle GPS systems."

Oh heck yes.

Somewhere in the back of my mind I'm recollecting having made a comparison between distances as indicated by a GPS and <something else>.  I think I was comparing my automotive-oriented GPS to Google Maps, maybe, and they were pretty close.  At other times I've compared the speed indicated by the GPS with the speed on my car's dashboard- also very comparable.

In all, the discrepancy doesn't worry me.  I just found it curious that it would be as much as three percent.

I just bring along my old GPS unit as a spare.

I calibrate my old-school wired cyclocomputer using highway mile markers. My calibration reads 0.1% low when the front tire is pumped to 70 psi, and 0.1% high 10 days later. It's totally repeatable.

I'm bummed that it is now hard to buy a conventional cyclocomputer with altimeter. Nowadays the main options are GPS cyclocomputers with a pressure altimeter and a wireless remote speed sensor. For my needs it seems like a giant step backwards to go from a wired cyclocomputer that runs for a year on one CR2032 button cell, to a GPS cyclocomputer that needs to be charged every week, PLUS a speed sensor battery that needs to be replaced twice a year. I don't want to store a track or upload routes. I just want to know altitude, gradient, total ascent, and total descent. It's hard to get that now without a bunch of other claptrap that drastically increases power consumption and significantly increases size, weight, and cost.

This is my ideal cyclocomputer, a 4 year old VDO M4.1 that I imported from MEC in Canada because there was no U.S. distributor. The wireless model was more common but my wired model works better with a recumbent bike.

Like most cyclocomputers it runs for a year on a CR2032 cell, but this one has a full suite of altitude functions: altitude, gradient, total ascent, total descent, max altitude, min altitude, max grade, min grade, and accumulated climbing (like an odometer for climbing).

It's a shame I can't buy an identical or better cyclocomputer now. The closest replacement now is a $40 Sigma BC-14.16 which has fewer altitude functions and a crude 2-line display. GPS gizmos have killed the market for top quality altimeter cyclocomputers.

My relationship with bike computers over the years has been of the love/hate variety.  I view them as data generators; when I have no particular need of / use for their data I don't bother with them.  And, when selecting a replacement- an inevitable necessity, every so often - I'm quite particular.

Features I want: current speed, trip ride mileage, total mileage, time of day, auto-start/auto-pause.
Features I cannot escape but don't care about: ride time, average pace.
Features I will not pay extra to get: wireless sensor, bluetooth connection, elevation change, cadence, heart rate (*), calories used, current air temperature, ATM PIN, turn my lights on at home from anywhere in the world, remind me what groceries I'm low on, auto-start my car, find my perfect match for a date/companion/life partner, blar blar blar.

Although they merely substitute one mode of failure for another, I don't particularly favor wireless pickups.  A broken wire is obvious (usually) whereas dead batteries in the sender unit only make themselves known when you actually try to use the thing.  And batteries are guaranteed to die, where wires may not break (though they usually seem to, after a while).

I've gone years at a time without using a bike computer, and not only not missed it but was actually pleased at its absence.  It's one less thing to worry about, fuss over, or be distracted by.  My focus can be entirely on enjoying the ride, and not at all concerned about whether I am "keeping up" any given pace.

Having never had a bike computer that gave me elevation / ascent / descent data, the way I keep track of elevation change is quite simple: I'm either climbing, or I've reached the top of the hill, or I'm descending, or I've reached the bottom of the hill, or the road seems basically flat.  If I'm climbing, I don't need a device to tell me the grade: my legs and lungs are plenty adequate to tell me whether I can manage it or not.

That said, I will admit that having the total climbing and max gradient numbers from my training rides may be useful, as I try to gauge where I am in my preparation and fitness relative to what I think might be The Real Deal when I'm on the road.  Then again, since I have no actual experience to use as a frame of reference, I may be deluding myself with visions of adequacy while catastrophically under-training, or I could easily be wildly and needlessly panicking myself.  Only time will tell which, if either, of those might be true; no amount of data will change the facts.

It's only when I am following a printed cue sheet that ride distance really becomes a factor.  If I know that I have X miles to go before the next turn, which lies at Y total miles into the ride, it's nice to have a relatively reliable indicator of when I might reasonably expect to arrive at that point.  Other than that, and for the purpose of logging miles for the sake of curiosity and later recall, there's not a lot of value in it for me.  And more than once when there's been a difference between cue sheet miles and indicated miles, it's been seriously disheartening that the end of the day's ride hasn't materialized as promised.

Since I'll be equipped with a cycle computer, a dedicated bike GPS, and a GPS/navigation app on my phone, I'll generate far more data on this summer's tour, and in the remaining months before I set off, than I'll ever need or want.  I recognize that to be a choice I've made, so I have no call to complain about it.  It is what it is.

Yes, there's something to be said for redundancy and backups.

When in the car, I've noticed that my Garmin auto (well okay, technically motorcycle) GPS makes different routing choices than Google Maps.  I usually follow Google and let the Garmin play catch-up but it's nice to have it along.  On many occasions my phone has overheated and shut down either Google Maps or the entire device, in an attempt at self-preservation.  In such circumstances I've been plenty glad to have the other GPS as a backup.

It'll be the same thing this summer: Ride With GPS will call the dance, and my bike Garmin will have to follow that lead.