The design is the most important thing
What you use for a finish on a paddle is a distant second at best. More importantly, it largely comes down to personal preference, rather than any important differences. Here's the basic scoop on finishes:
Oil
The main reasons that people choose oil finishes are twofold:
1 - They like the feel. Oil finishes (pure oils, not faux-oil finishing products) cannot be built up to any significant thickness. They produce a soft, matte or satin finish. Consequently, an oiled paddle still feels like wood. Many people, myself included, prefer the feel of an oiled paddle.
2- They like the look. The matte/satin finish is visually appealing because it looks like wood, not like wood that has been coated with something.
Other plusses for oil finishes:
- They are easy to maintain. You simply clean off the paddle, wipe on more oil, then wipe off the excess after 5 minutes or so. OTOH, being relatively soft, they also require more frequent maintenance than other finishes.
- They allow the wood to breathe to an extent, so that any water that gets through the finish can get back out again.
The most common types of oil finishes are:
- Raw linseed oil. This is not generally a good finish for paddles or much of anything else, as it takes weeks or even months to cure. I don't have that much patience nor do I see any advantage to raw linseed oil.
- Boiled linseed oil. This is by far the most common oil product on the market. It's not actually boiled, but it contains heavy metal driers (cobalt) that cause it to cure much more rapidly than its raw cousin. Applying one coat per day is pretty standard. One downside to either linseed oil is that mold and mildew feed on them, which is why they're not a good finish for items that stay wet for long periods of time, like SOF kayak frames. For paddles, this isn't typically a problem.
- Tung oil. Pure tung oil is good wood finish, but you have to read the label. There are a lot of "Tung Oil Finish" products on the market that contain little or no tung oil (they're wiping varnishes - see below). I prefer to use tung oil for paddle finishes, but the functional differences between it and boiled linseed oil are not significant.
Varnish
Most commercial paddles are finished with some form of varnish, either spar varnish or polyurethane (polyurethane is simply varnish made with synthetic oils and resins). It creates a hard film finish that can be very durable. Typically, gloss varnishes are used, which gives the paddle a plastic look and a rather slippery feel. Satin and matte varnishes are available, but with use they get polished in the grip areas and become just as slippery as gloss varnish. You can "renew" the finish with Scotchbrite, steel wool or 600 grit sandpaper, but it just gets polished again with use.
Being a completely waterproof film finish, if it becomes damaged and water wicks into the wood, it becomes trapped and the wood will likely become discolored in that area unless you sand it down, let it dry and refinish it. Rot is possible, but realistically, it's not likely on a paddle.
Epoxy
Epoxy is a hard film finish, but if you wipe it on and off like oil, it can produce a more natural look and feel than gloss varnish. However, epoxy is UV sensitive, so it really should be covered with varnish to protect it, which is what you usually see on commercial paddles. It hardens the surface of the wood substantially and may stiffen the paddle to a noticeable degree. If applied thickly and/or coated with varnish, all the caveats regarding water entrapment apply. As with varnish, it's more involved to maintain it, especially with epoxy/varnish finishes.
There is a class of products called "wiping varnish", which are mainly varnishes with a lot of thinner added (mineral spirits), but sometimes with oils added as well. Most "tung oil finish", "Danish oil" and "Teak oil" finishes fall into this category. Depending on their composition, they will be more like varnish or more like oil. These are generally a very poor value, as you pay a premium price for a product that's mostly cheap thinner.
To further muddy the waters, there is class of oil finishes known as "polymerized oils". Depending on the degree of polymerization, they will behave like oil or varnish. Birchwood Casey Tru-Oil is a good example of a highly polymerized oil that acts like varnish. Polymerized oils are not that common and you're only likely to see them in specialty woodworking stores. Some are marketed as "natural" or "green" finishes, since they don't contain petroleum-based thinners and heavy metal driers, like many other products.
My personal preference in paddle finishes is a 50:50 blend of varnish and pure tung oil. It retains the look, feel and ease of application/maintenance of oil finishes with considerably better durability. I epoxy coat the tips of my paddles for durability and since it's a sacrificial coating, it wears down and requires re-coating before UV damage becomes an issue. I find this to be the best application for these products, at least for my personal use.
I guess the bottom line is that different finishes are appropriate for different uses and paddlers. If someone tells you that a particular product or procedure is "the only way to finish a paddle", turn around and walk away.