propeller vs paddlewheel

The tug of war you are talking about was between HMS Rattler (fitted with propellers) and HMS Alecto (a paddle wheeler). Off the top of my head other than the propulsion system both ships were the same size and had engines producing the same amount of power.

This trial has been reproduced using RC models, with the same results - the propeller driven ship is quite capable of pulling the paddle wheeler backwards without trouble, even on a flat 'sea'.

Paddle wheels have a number of problems, especially on open seas. The design means that the engine has to be placed fairly high up, which reduces stability. The paddles themselves are mostly out of the water, so much of the engines power is lost and last of all if the ship is rolling one of the paddles is going to be lifted at least partly out of the water resulting in imbalanced thrust and strain on the propulsion system. (In extreme weather you can even end up with neither of the paddles being in the water).

For military ships of the period paddle wheels had two additional and related problems. The paddles size meant that you couldn't put guns in the middle of the ship, while at the same time were large and inviting targets for enemy gunners. So paddle wheelers were not considered practical warships, as they were poorly armed, highly vulnerable to being crippled if shot at, used a LOT of fuel and had problems with relatively mild seas. You couldn't even realistically use sails to overcome some of these problems as the paddles would provide a lot of drag.

Propellers could place the engine lower down, making them more stable, and were more efficient since all of the energy going into the propulsion system was being used to move the water and hence the ship. They also don't have to worry about heavy seas, since the propeller is unlikely to come out of the water and even if it does this would not create strain on the system.

Yes, and its actually quite simple.
(Or so I hear... this is anecdotal evidence from someone who should have known, My Dad, who was Merchant Navy before I was born...)
In heavy seas, the paddle-wheel would not be always at the right height in the water to provide maximum thrust.
The prop, however, would be under-water all of the time... if it was ever out of the water, you had slightly more problems going on than the ability to push?....

It had nothing to do with size, engine placement, exposure of the paddles to damage, but all to do with in rough seas, the prop was the winner, and that was the end of that.
Sure, paddles are nice, on a flat surface mill pond, but anything else, they are a nightmare.
Props are a one-size-fits-all solution for all conditions.

Modern technology...?...
Well, you would hope to impro9ve, eventually...
I was comparing only two technologies, and in that comparison, Props won, but that was centuries old comparison, and we have re-invented the wheel since then.

Not that I know a lot about this, but I'm guessing paddles would also need a lot more attention and repair than a prop. From pure experience, the larger and more complex you make something, the more that can and will go wrong and a paddle system is conciderably larger than a prop system. Most of a prop's system is also inside the ship, except for the actual prop itself and a very small portion of the drive shaft, so I'm guessing it's a lot easier to maintain and repair than the paddle system where most of it is on the outside.

Props can do more self damage, as they are the unseen....
The wheel under the back can be damaged by Cavitation even on the easiest day, it can collide with water born objects, it can get a little loose, and wear, and mostly, its only looked at when something goes wrong.
I have seen props being used with a blade missing... "Get home" principle, but how much damage does that do?...