People are always shocked to hear that I whip cream, beat egg whites, and make mayonnaise by hand, as if it's a superhuman feat not to be attempted by ordinary folk. But the truth is that I'm lazy, and I'd rather spend three minutes snapping a whisk around a bowl than deal with an electric appliance. The whisk is lightweight, always within reach, and cleans up quickly. Blenders and food processors have to be taken out of a cabinet and set up, and cleaning them is more of a nuisance. I find it a pain to get the immersion blender, too—it’s in a cabinet, and the cord is coiled around it, so I have to uncoil it, plug it in, and then, once I'm finished using it, clean its blades, dry it, and store it away. All that, just so I don't have to move my arm? No thanks.
But I don't just use a whisk to avoid dealing with cumbersome appliances. There are real reasons to opt for the manual process, just as there are sometimes reasons not to. Whipped cream, for instance, is a heck of a lot harder to overbeat when you do it by hand—its transition from a liquid to a foam is more gradual and your margin of error is therefore greater. Egg whites, on the other hand, often benefit from the extra power of a stand mixer, making the case for beating them by hand less strong.
As for mayo, it all depends on what you want. Choosing between the whisk and the blender is less about whichever you feel is more convenient and more about what qualities you want the finished sauce to have. A mayo whisked by hand is saucier and glossier, while one emulsified with the help of a machine is thicker and creamier. Each has its place.
To demonstrate my point, I made two batches of mayonnaise using the exact same ingredients and ratios. The only difference: I whisked one together and used an immersion blender for the other. The whisked one flowed slowly off a spoon, while the blended one had to be shaken off in dollops.
The differences go beyond texture, since the method used has an impact on flavor, as well. My hand-whisked mayo tasted brighter and more lemony, even though it had the exact same amount of lemon juice (from the same lemon) as the blended batch.
The whisked one was also less bitter. There are two possible explanations for this. First, the mayo I made contained a 50/50 mix of olive oil and neutral grape-seed oil. High-speed blending can sometimes make olive oil taste more bitter, affecting the flavor of the mayo. The other possibility is the garlic. In the hand-whisked batch, I added minced garlic. Whisking isn't capable of breaking the garlic down any further, but a blender is. The minced garlic I added to the blended batch was processed even more finely by the spinning blades, releasing more of garlic's bitter flavor compounds. The effect of mincing garlic more thoroughly can be a profound one, as I found in my garlic mincing tests a few years ago. Of course, this doesn't have to be an either/or; the bitterness of my blended mayo batch may well be a consequence of both these factors.
The differences between these two versions of mayo, both in terms of texture and flavor, are due to how thoroughly an emulsion is formed. When you make mayonnaise, what you're doing is suspending tiny droplets of oil in a water medium (the "water" being an acid like lemon juice or vinegar) with the help of an emulsifier; in the case of mayonnaise, the emulsifier is an egg yolk. The tinier the drops of oil, the thicker and more opaque the mayo becomes. Since whisking by hand is less powerful than an electric blender, it can't form droplets as small. Hence the glossier, saucier result with a whisk.
So, which method should you use? It depends on what you want. I prefer a hand-whisked mayonnaise when I want to use the emulsion more like a sauce to go with a roasted or grilled fish, say, or as a dip for vegetables. If I'm using the mayo more as a spread, such as on a sandwich, I'm more likely to blend it. The thicker mayo will stay put better and be less prone to squishing out the sides of the sandwich when you take a bite.
The next time you make mayo, think about which effect you'd rather have before deciding which tool to use. You'll be a better cook as a result.