My own observations seem like one side is for stabilization while the other does the fine movements. But people also typically have structure assymetries - I've got a distinctly larger and more defined peak on my left bicep vs my right - they differ by 1/2". My left quad tends to look better than my right. My left foot is 1/2 size bigger than the right and my right hand is bigger than my left. But however it all comes together, in isolation you will probably notice differences. I think its good to exercise both in tandem as well as alternating to force the weak side to be able to work on its own.
I probably wouldn't go too far out of my way to change my training to accommodate the weaker side, but I'd introduce some alternating exercises simply to allow each to be able to operate on its own. An easy example is a barbell press vs DB press. Or use of cables vs machines. For legs it can be a little harder, but particularly when I'm doing some sort of active recovery, I'd mix in something like rep sets on the leg press. Its hard to do drop sets if you don't have someone w/ you to strip the weights for each set, but you can get a lot of work out of drop reps - alternating leg presses - 20 reps per side, then 15 reps per side, the 12 reps per side, 10 reps per side ... until you're down to min. Other options to integrate into your routine are walking lunges or reverse lunges w/ a smith or barbell.