After the rise of Jessica Fox and many other men and women in C1 slalom. I am a firm believer in switching sides whenever you think it is better.
We are drifting away from flat water freestyle, but switching hands is an important functional paddling issue. Lennart's reference is that many top whitewater slalom racers now switch hands at appropriate times to get the best leverage, force and efficiency for the next upcoming slalom gate, given the changing water architecture. In days past, we learned "never" to switch hands in whitewater.
I changed my mind about that in 1987 when paddling on Section IV of the Chattooga River with Nolan Whitesell. Nolan, to our surprise, would tactically switch hands in the flats between class 3, 4 and 5 rapids if they were significantly "lefty" or "righty" rapids. Or he'd switch sides in an eddy in the middle of a long rapid if the next upcoming move was more efficiently paddled lefty or righty. He could paddle any rapid on either side—and, of course, he had cross strokes from either side—but some rapids and moves therein are definitely easier if you paddle them lefty or righty. Nolan could also roll with a paddle or just with his hands lefty or righty, and do both on-side and cross-side rolls with paddle or hands.
Most non-freestyle flat water paddling is simply forward stroking. I switch sides occasionally when doing that, especially as my body has aged, simply to exercise the muscles on both sides of my body and to avoid muscle/tendon strains (as I currently now have in my right forearm) from correcting too much on one side only. I also paddle hit & switch from my kneeling position in certain situations when I need more velocity, such as during long crossings in winds or paddling up-current in rivers.
Finally, to get back to the question of the difference between functional freestyle flat water paddling and just regular tripping flat water paddling, I agree with
@tketcham. When day paddling, just for fun and to break up the monotony, I'll deliberately try using different moves to get around trees, branches, and other obstacles. I'll also deliberately try different ways to go around a bend on a twisty stream.
For example, if I'm faced with making a sharp turn around an off-side bend, my earliest (childhood) instinct would have been to use a sweep stroke. But that might cause me to slide to the outside too much in strong currents. Later, mainly from whitewater training in the early 80s, I learned to make the turn with a cross-draw followed by cross-forward strokes. About the same time, I learned the benefit of heeling the hull to the inside or outside of the turn, depending on the "preference" of the hull's architecture. Finally, about 16 years ago at freestyle instructionals, I learned the highly leveraged benefits of making that off-side turn with a wedge or inverted wedge (aka bow jam).
There is a similar panoply of on-side turning manuevers, which are fun to alternate, play with, and hence practice. I find that each one can have slightly more or less benefits in certain situations.
To me, variety is the spice of canoeing life—namely, variety with different hulls, different paddle shapes, different paddling sides, different paddling techniques, different moves, and different waters in different places.
