The only difference is in the levers/trigger guards (the No. 3 lacks one). And the stocks, if we're comparing factory issued guns.
Many a custom gun has been made using the No. 3 action and I have a nice one too. The No. 3 is, or used to be, a lot cheaper to obtain, but is every bit as strong and versatile as a No. One, so that's why it was used so often for custom projects. You can alter the lever into a very elegant design, whereas the No. One lever is pretty much what it is.
The stocks on the #3s were clunky with a straight grip, plastic butt plate and a stepped forend with a 10/22 style barrel band. The lever was S-shaped and the barrel had a different taper than the #1s. The only way a #3 looks good is with #1 wood and that causes issues with the lever. It can be reshaped to resemble a High Wall when using #1 wood.
The #1 is the most beautiful single shot ever made here without any cosmetic changes.
Depends. To me "customizing" means doing some tweaks and upgrades. In that case, No.1 for certain because it's already pretty nice. For a full build using only the action, it would depend on the final destination. The simple Winchester-style lever on No. 3s is quite handsome on the right rifle, just like the rifles that inspired it.
You didn't say what you want to build, but there are very nice Rugers and Brownings to be had, NIB, or nearly so at prices far below any custom. My favorites in deer-capable ones are the Low Wall .243s and .260s. Figger $1500-$2000 for nice ones. Lots of special-run Rugers have been made too that are stunning. Just takes a bit of patience.
The No. 1 also has a barrel rib for scope mounting.
If you want to work the existing stock then either one can be turned into an interesting custom depending on what you like. The No.1 can be very elegant but the No. 3 can be made up into a very trim little rifle as well. The No. 3's generally had much plainer wood back when Ruger was putting decent wood on their No. 1. Nowadays the wood in No. 1's isn't much to write home about anyway. If you want to restock with new blanks then the sky's the limit on how fancy you can go, Then it would be more a matter of which lever style you like and whether or not you want a quarter rib - although even that can be removed if desired.
Can't display the pics directly here since they are proprietary to his site but this guy reworks both models into very nice designs, either by modifying the factory stocks and leaving the barreled actions alone or he can strip out everything but the receiver and start from there.
Here's one I just got on Gunbroker, mostly because of the price and the fact that it came with custom English Walnut, a $300 Kepplinger set trigger. a nicely altered lever for pistol grip stocks and a nice vintage 6.5-20X Simmons silhouette scope.
It's getting different wood, a different lever and a different barrel and will become a 32-20, something I've always wanted in a Ruger single shot.
as everyone else pointed out some of the differences but one not mentioned is that the number three has a ball and detent assembly in the front of the lever to keep it closed. that is one of the reasons that the lever can almost be altered easily any way you like. I have seen them from a high wall look to really ornate schutzen look lever.
One might note that the factory lever on the No.3 can jam the fingers on shooting a fairly light rifle in a heavier caliber. Not everyone is bothered by such, but I got rid of some Browning 78's because of that, fingers were always getting smashed.
I am currently working on fitting new wood to a #3. I was initially worried about the lever, but with the stock bolted on, seems it may actually work for me without altering the lever.
Problem now is jumping off the fence on the barrel. It is 223 and shoots real nice, but the contour will look like crap with the new wood. I want to put on a 257 Rob barrel on it, but I am stalling because part of me wants to leave it as is. Got to make a decision before fitting the forearm.
I put No. 1 wood on No. 3 action with no lever issues. I prefer the looks of the No. 3 lever but am not fond of the butt stock and forearm. I have the original No. 3 scope blocks that take the Ruger Rings but have not used them yet.
If I were building a custom rifle with a new barrel and stock either one would be just fine.
This is my Ruger No 3 that has been modified. The reason I modified mine was at the time Ruger No 1's in my area were impossible to find and I lucked into this No 3. This rifle has a Ruger No 1 stock set and an added 1/4 rib with a band type front sight. The lever was also modified to make it easier to use with a pistol grip stock. It is very handy and shoots great.