That looks very like a tucker equitation endurance type saddle, I'm intrigued now
The synthetic one looks like a pleasure or trail saddle, its difficult to tell which without hands on.
-Pleasure saddles generally have a low swell fork and horn, more there for decoration than function (you can't rope off them) and a high-ish cantle, they have neutral or fractionally forward set stirrup bars and rarely have a rear cinch. Within this category you also get equitation saddles, which are similar but sit you in a much more classical balanced position with you legs well under you.
-Reining saddles generally give you a flatter more open seat and have a minimal swell fork, with a reasonable cantle and the stirrup bars are set slightly forwards for when you do those big stops and a short forward swept horn meant to stay out of your way (again you can't rope off it) again they rarely have a rear cinch.
-Cutting saddles have a tall thin horn, meant to be held on to with your fist for those super quick cutting movement, they have swell forks which are quite high and stirrup bars set well under you and the fenders are often rough out to get you more stick-ability and a very open flat seat which is normally padded.
-Barrel saddles generally have a narrower twist, a high cantle with a padded seat and enough horn to hang on to (again not for roping) but its not as tall as a cutting horn, they rarely have a rear cinch.
-Trail saddles are the GP of the western world, they generally have a middle of the road seat which is padded, with a reasonable cantle and a stocky horn which you can do light roping on, they are set up for a comfortable ride and often have a rear cinch for stability and are normally the lightest next to a barrel saddle.
-Roping saddles have a much thicker horn and the swells are often thicker too to cope with heavy roping, they should always have a rear cinch to prevent the back of the saddle lifting when you are roping something heavy, these are generally much heavier than any other type of saddle, the seat generally offers more security with a medium high cantle and may or may not be padded.
-Wade saddles are ropers but they are in a bit of a class of their own in terms of style, these are more of the old timer ranch saddles, they are heavy, good for roping and very comfortable to spend all day in the saddle even though they normally have a completely unpadded seat (I find these the most comfortable of all). These normally have A forks and a large horn.
So there are many different styles of western saddles and finding the one that suits you can take a minute, just like with English saddles, there is also massive variance between brands. Most western saddles also have quite a wide twist, unless it is specifically a women's model, barrel racing is generally a women's spot so those are generally narrower.