Increased power density is indeed one allure of wave energy vs wind. In the same way that wind is concentrated solar energy, studying the mechanisms and origins of waves show they are essentially a form of concentrated wind energy. So if the difficult design challenges are solved, there is potential for greater energy per unit of structural material which some believe could be associated with competitive cost of energy.
Other allures of wave energy include that it is fairly consistent, with some seasonal variations, and its decoupled from solar or wind which decreases the chance that all renewables resources aren't producing at the same time. Of course this is talking about wave energy where oscillations occur with a period of approximately 4-18 seconds rather than tidal with several periods per day, and for these periods of time flow is essentially unidirectional. Tidal energy, like the link in this thread, often uses underwater turbine but wave energy devices rarely use this design.
There is a wide range of ideas, most are at small scale, early stage, and no winner has emerged the same way that wind energy has seen the 3 bladed turbine win out.
Wave energy devices exist using almost any type of power take off mechanism - direct drive electric generators, linear generators, compressed air, and hydraulic actuators. Some may even use propellers in water but that's not as common since direction is always changing.
But the power generated by a turbine is proportional to the cube of the air/fluid velocity..... and the wind normally moves much faster than tidal flow.
It's also worth noting that when dealing with centrifugal devices, the realized power never actually follows the power law. It's usually degraded quite a bit due to linear losses. A common power used in air systems is 2.1 rather than the cube; using that value, the airspeed must be almost 24x the water velocity. I've never personally seen anything greater than 2.7 (the higher numbers, IMO, tend to be used as a sales pitch rather than an engineering decision), but again, I don't work in these particular systems.