The biggest difference between these “carbs” is that the McDonald’s fries have far more fat per gram of carbohydrate. For someone who is training intensely but watching total calorie intake for body fat management the sweet potato is far superior in that regard.
Also vitamins, minerals, etc.
More pre-processed food sources will likely be more easily converted to their constituent monomers. Probably affects satiety more than training per se.
Most "carbs" are high fat as well.
The short answer is that no, the complicated moral psychology we impart to food is not something your body is capable of perceiving.
The long answer is that there are various characteristics to food that inform this moral psychology, some of which are more relevant than others.
The levels of insulin that a given carbohydrate produces is something of a scam by the fitness and nutrition industry. All carbohydrates (except for fiber and some sugar alcohols) require the same amount of insulin to be processed by the body. "Quicker" (high GI) carbs might cause a sharper rise in blood sugar than "slower" carbs, but this is mostly irrelevant (you would be hard pressed to get your blood sugar and insulin levels to fluctuate more than a type I diabetic with extremely tight control, and with said tight control those people don't experience any deleterious effects).
The characteristic of carbs that matters more than pretty much any other is satiety. Carbs contain more or less fat, fiber, and protein, which causes them to be digested slower or faster, and to be available as energy at different times and rates, and hence produce different sensations. A "bad" processed, high GI carb is not to be avoided because it "spikes your insulin", but because it moves through your stomach and into storage quickly, meaning you will be hungry and tired even though the same amount of calories were available to you. You've heard Rip tell people to avoid sugar. This is not because sugar carbs make you fatter than non-sugar carbs, but because you can eat sugar with pretty much no limit, which means you'll eat in excess of your body's needs. Rob calls them "black hole" foods: foods that you could eat an unlimited amount of given the opportunity.
If you weighed and tracked your food perfectly, there wouldn't be a big difference physiologically between you getting all your carbs from white rice or from pixie sticks. The difference is with the pixie stick diet you're probably going to be a lot hungrier and more pissed off than if you got to eat an occasional meal.
This is why processed foods are "bad": not because they put shit *in* them (which is a quirk of psychology: humans are paranoid about bad stuff being in food when it is "unclean." It's why every other religion has dietary prohibitions, but very few have dietary proscriptions) but because processing strips shit *out* of them. Fiber, protein, vitamins, you name it. You can eat all the saturated fats and seed oils that you want, the problem comes when you're eating a huge amount of those because none of your food has fiber so you never feel full, or when you aren't moving all day because you're in a permanent sugar crash.
If we get rid of the difference in fat content, vitamins and minerals, etc. the answer is still yes. Different carbohydrates are turned into blood sugar at different rates. This rate can be measured, and is expressed as a glycemic index (GI). A low GI means that eating that food will raise your blood sugar slowly, and vice versa.
Whether a low or high GI food is "better for you" depends entirely on why you are consuming the food, but in general it's a good idea to consume more low GI foods most of the time.