That's interesting. All my pre-Weller life, I used the damp sponge in the little iron holder, but since switching to the Weller station, I've been liking the copper coils much more. The coils seem (to me anyway) to remove the excess solder without wiping off too much (thus keeping the tip lightly tinned), and doesn't cool the tip off as much as the sponge does, which helps keep the work moving.
Not declaring one right or wrong of course (to each his/her own); just interesting to hear about someone switching from one to the other.
A couple nights ago, I was literally switching back and forth between both as I was soldering. Sometimes you swipe on the sponge and you still have a gob of solder that doesn't want to vacate, stuff the tip in the copper-coily and da - da, da da dahhhhh satisfaction.
First, keep the tip's temperature in check. No matter what, the tip will be oxidizing, all the time. A hotter tip will oxidize faster. This is a big advantage of regulated irons - you can set the temp appropriately and the tip won't just sit there cooking itself when it's idle because the iron is dumb enough that it doesn't know the difference. So, first things first, adjust the temp so that it's hot enough to be useful, but no hotter. A high wattage iron is nice here, since it can dump more heat into the tip to keep the temp correct even when soldering big parts. Thus the magic combination - high wattage to move heat quickly, but regulated to keep itself from burning the tip up.
Oxidation prevents heat flow. So, you want your tip to be freshly cleaned immediately before each use. A tip with a thick blob like layer of molten solder on it will be easier to wipe clean, since you can wipe away the oxidation with the excess solder (the oxidation will be on the surface of the molten blob).
To achieve that, my routine is to turn the iron on, get it hot, and put a big blob of solder on the tip. Then leave it in it's holder until I need it. As I pull it out for each joint, I wipe it clean on the brass cleaner, then use it. After use, touch the solder to the tip to put another blob on it. Then it goes back in the holder for next time. To answer your question - ideally you're maintaining a little thickness of liquid solder while you're working, and a big thickness while the iron is idle, so it's easy to clean right before use on your next joint. So the rhythm is: clean, use, tin, store.
That's the key: Any time the iron is idle, it has a blob on it. Any time it's in use, the blob has been freshly wiped off so when you're working, you have a 100% "fresh" clean tip with a thin layer of very clean solder on it.
Then, if need be for certain parts or certain techniques, you can add more solder to the tip as required. Or if you're populating a big board and doing a lot of joints at once, like you're tinning a bunch of SMT pads prior to populating, you might pause every dozen or so to tin it and wipe it clean mid-use. But the basic rhythm of clean, use, tin, store stays the same all the time in addition to those special cases.
That is the best explanation of why you should tin the tip between working with it I've ever come across, thank you.
Most explanations simply state tinning "protects the tip...", some might include "...from oxidation". Your explanation of "heat flow" and "wipe away the oxidation with the excess solder (the oxidation will be on the surface of the molten blob)" goes beyond what to do and offers more than a tacit why.
Another tip for newbs like myself — copper pads are cheap, so when you stuff your tip in to the copper pad to clean it, and the tip comes out of the copper with more solder on than when you put it in ... it's time to get a new pad.
Same for the sponge. So I keep a spare of both the copper-cleaner pad and a spare sponge. After replacing (well, usually just before) I purchase a new backup.
I thought I saw somebody mention using the copper brillo or Scotchbrite for cleaning old PCBs / Vero / Perf board — maybe it was mentioned in another thread? Anyway...
I've mentioned this before in other threads, but thought it may prove beneficial to mention it here, too:
My solution for old oxidised PCBs / Vero / Perf board is a trick I transposed from my days working in a bike shop, from before the days of disc-brakes on bikes...
That ☝︎, dear solder-slingers, is a rim being cleaned of brake-dust and road-grime; what is pictured in hand, note, is the nub of one of these:
It turns old oxidised dull-brown copper into shiny bright newly-minted-penny-like copper.
I had a tiny piece of old vero that was beyond dull, but as I was still waiting shipment of a new batch of vero, so this vero-snippet was all I had to use — I knew the copper was badly oxidised, but tried to use it anyway and the solder wasn't flowing like it should, wasn't sticking. It struck me I could use my Rim Rubber (stop your snickering, I'm trying to be serious here)...yes, rim rubber, and so I tested it on a corner of the vero. Well, it worked so well that during my next shift at the bike shop I bought one of these Abrasive Erasers solely for soldering.
It works on PCBs' shiny silver bits, too, without erasing the solder mask or the lettering (I suppose if you rubbed too hard and too long, you might erase the PCB's silk-screening and make a Genie appear). After erasing the oxidation, wipe the surface clean of any eraser residue.
There are other brands other than Mavic, of course...
If you're a pro solder-jockey, it's unlikely any of your stock of PCBs/Vero/Perf will be around long enough to tarnish.
If you're a hobbyist like me, however, one who hoards years-old PCBs, Perf and Vero, then one of these abrasive-erasers is ideal, and should last several years if not decades before it's down to the nub. It's not like the stuff we solder is subjected to the same level of suffering the rim of a bicycle-wheel is, abused by brake pads and road/trail grit... SO!
Right now, run! Don't walk, RUN to your nearest Independent Bicycle Dealer and purchase your very own
Gomme Abrasive/Abrasive Rubber/
Felgenreinigungsgummi — Polish those PCB pads to a shine, make your perf project perfect, ensure that solder flows efficiently and verily to your vero.
Batteries not included, some settling may occur during shipping and handling, this offer not valid in some areas, always have adult supervision when operating product, do not operate product while under the influence of drugs or alcohol, objects under the magnifying glass will appear closer than they actually are, your mileage may vary, put the rubber down and keep the shiny side up. 
I was both floored by the quality and ease of use of it, and mentally kicking myself for using cheap irons for the last 30+ years. 
To say the least, I no longer consider a good quality soldering station a splurge; As with any other tool purchase, sometimes you get what you pay for. This one has been well worth it. 
