As far as feed, farrier, and vet, there isn’t a drastic difference in cost between a young horse and an adult show horse.
Nutritional requirements for a growing horse vs. a mature horse in work are fairly similar, so there shouldn’t be a drastic difference in feed cost. However, a young horse can be quite at risk to developmental orthopedic disease if not fed properly. One can’t “skimp” with low quality nutrition on a growing animal like he or she may be able to on a mature horse. (Which I’m not saying you do, just emphasizing the importance of nutrition)
Farrier costs should be the same. Young horses should be trimmed on a regular 4-6 week schedule just like mature horses. If you generally have horses in shoes, you may save a little money on shoeing costs, as it’s unlikely to have a reason to shoe an unbroke youngster.
Vet costs vary greatly. However, if you have the vet give your vaccinations, you may have an increased cost that first year for the initial vaccination booster series. Most breeders begin their foals’ first vaccinations between 4-6 months of age, so depending on when you acquire the foal and what the breeder did, you may or may not have multiple vaccinations to give. You also have to consider things like gelding an intact male, wolf teeth extractions, etc. Young horses do tend to be more prone to catch whatever bug is going around or find trouble, so there’s a slightly increased risk of “surprise” vet calls.
You also may have the expense of horse supplies for the growing animal: halters, blankets, fly masks, etc.
I raised my mare from birth. I loved it and would do it again. However, I will say my own priorities as a rider greatly changed from my mare’s birth to now. When she was born, I was actively competing and she was going to be my show horse. These days, I’m at a different place in my life and have lost interest in competition. I’d be much better off with a pokey old trail horse to putz around on… which my mare is most definitely not! I feel bad she’s wasting away as a backyard pet who packs me around for my entertainment, but at the same time, she is like a child to me and I would never “replace” her.