I think Bill gave a very good explanation as to why shooting on a tripod is actually easier that trying to accomplish the same hand held. My hand held shots, too, tend to be impulsive and not well thought out. Nor is the composition always what it should have been.
From a technical point of view (sharpness) I don't think there is a clear cut answer. It depends on the focal length, shutter speed, distance to subject, sensor pixel count, and etc.
I once tried to answer this question mathematically, as a function of the standard 1/FL rule of thumb. I set up a target (a test chart I have used for many years and know well in terms of "what is sharp(est)"). I shot a careful tripod shot, using Mup and all the technique I would use to get a perfect picture.
Then I shot the image hand held, taking several images at each of various shutter speeds.
I was actually surprised at the results. I was able to get reasonably consistently sharp shots at "only" 3-4x focal length, so at 50mm that would 1/150 to 1/200s. Before doing this I thought I would have to go higher.
When the D800 was announced there was an ensuing huge argument over the issue of hand holding that camera.
And I want to say that there is also an argument over what sensor spec determines the "hand hold-ability" of a digital camera- is it the pixel count or the pixel density. I can show you mathematically that it is pixel count, not sensor density, and the size of the sensor containing those pixels has no bearing on that.
A tiny 12mpx P&S sensor is no easier or harder to hand hold than a medium format sensor of the same pixel density (all else equal such as shutter speed). All else equal also assumes same field of view, regardless of focal length.
I didn't have a 36 mpx sensor to do the test (nor did anyone else at the time ). But I used what I had, just to see the state of affairs with the sensors available to me, which was my D700 and D300.
I only tested a short focal length- around 50-100mm, as I recall. I don't know if that would hold up to 200mm with something like my 70-200 VR because the lens weight starts to create issues that transcend the focal length. If my 50mm lens was the same weight and size as my 70-200 then that would help equalize things. (does that make sense?)
I do a lot of birding at 700mm with a 500/4 on a tripod and gimbal. I am amazed that there are times I can't seem to get a sharp shot even at 1/1000s. Not often, and it is difficult to analyze the reasons after the fact. Things like focus. Just to say that as focal length increases I think the issues are complicated and compounded and more difficult to sort out.
Good tripods are heavy. Even better tripods are made from carbon fiber (CF) and are not . And this is one argument for CF. it helps to take the portability aspect out of the decision.
I think some photography, such as street shooting, certain types of event shooting, and basic candid portraiture is usually best done hand held because those images depend on fast instants in time and require agility to capture those moments. Other types of photography are best done in a more studied and slow pace, and involve basically static subjects. You have to factor that into the question.