The Ruger No. 3 was a less expensive version of the No. 1. Cost reduction features included no checkering, less-highly-figured wood, a metal (later plastic) buttplate (ala the basic 10/22), a simpler actuating lever, less-highly polished metal and bluing (although that is debatable), and iron sights with no ramp or rings provided with the rifle (rings are provided with the No. 1). The basic action (except for the operating lever style) is essentially the same.
The No. 3 was introduced around 1973 and discontinued sometime in the 1980s. It was initially available in "classic" cartridges (including .45-70, .22 Hornet, and .30-40 Krag). Later chamberings included .375 Winchester (relatively rare), .223 Remington, and .44 Mag (also relatively rare, thought to be more rare than the .375 Winchester, and the subject of some discussion on this board). There are one or two oddballs out there...a variant of the No. 3 was chambered in .30 Carbine as a short barreled action contained inside a simulated Viper launcher designed to be used as a military trainer (the Viper project was killed and with it, the .30 Carbine No. 3 simulator - I worked on the Viper at General Dynamics 35 years ago on this project and I handled these simulators). I've heard that one .30 06 No. 3 was built as a prototype but never entered production, but that's hearsay. Other than the different chamberings, Ruger No. 3 rifles were only built as one model.
A Ruger No. 3 in .45-70.
Ruger No. 1 rifles have been built as heavy barreled varmint rifles (the 1V configuration), Alex Henry fore end models (the 1A, 1S, and 1H models), RSI models (with full length Mannlicher-style stocks), and "standard" models (lighter barrels and beavertail fore ends, usually designated as 1B models). There are several variations on the above, including many distributor specials. As far as calibers go, it might be easier to list the cartridges that have not been chambered for the Ruger No. 1.
A Ruger No. 1A in .30 06, purchased in 1976. Finding No. 1 Rugers with wood like this was not unusual in the mid-1970s.
Back in the day when I started collecting (in 1976), MSRP for a new No. 3 was $165, and you could buy them all day long for about $138. A new No. 1 in any configuration carried an MSRP of $265, and you could buy them all day long for about $235. (Those, indeed, were the good old days.) Today, used No. 3 Rugers go for around $650 and go up from there (there will be some folks who will post here that they can be had for "a lot" less; where this occurs, please point me toward the sellers). Used No. 1 Rugers start for just a bit more, usually around $700 to $750, assuming the rifle has the usual handling nicks and dings and it is not rare. MSRP for a new Ruger No. 1 is $1299, but the only retailers who sell them at that price are retailers who aren't selling anything. They can be had brand new for $900 to maybe $1050 or so. The Ruger No. 1 has many collectible variants, with some of the more rare configurations selling north of $1500 (sometimes well north of that number).
As far as accuracy goes, the No. 1 is perhaps more inherently accurate than the No. 3 because of the No. 3 rifle's barrel band. That's the theory. On open sighted rifles shooting the .45-70, I can't detect a difference. My No. 3 .22 Hornet is of modest accuracy (around 1.5 inches at 100 yards with 296 powder), but I don't buy these for accuracy. If accuracy was my criteria, I would buy a Savage bolt action. Either the No. 1 or the No. 3 will deliver acceptable accuracy for hunting (in my opinion; others will probably disagree).
In my opinion, the single shot Ruger rifles are the most elegant rifles ever offered as a normal production item. I think there are others here on this board that would agree with me.