That depends on how you homeschool and what you look for/find in private school.
Personally choosing one or the other should be more about the quality of education than price.
I homeschool ODS (oldest) for first grade, it cost about 300 for materials (some of which are reusable for DD later) If he were in a 'non home school' public or private I estimated the cost at about the same (not including tuition if aplicable) when you send a kid to another source to learn you have to get them there (and even if you do use the bus what about when they miss it?) you have to supply more than just one kid with the crayons, pencils, paper, notebooks, and all the other paraphanilia (my kid uses the same pencil he had in k-5, that NEVER happened when I taught in a school!) Not to mention all the 'class party stuff' nothing wrong wiht any of that, but 'public school' ain't free, private schools have (for the most part) the same 'stuff'.
Oh yeah don't forget all the TIME! time is money to many, and homeschooling takes about 2 hours of sitdown stuff a day, the rest is all well, fun, stuff we do in our free time. Just so happens that it is learning, and generally free

. As opposed to a school where a kid is locked up for hours at a time, and not much of the free time is spent learning, instead it is waiting a turn, in line, listening to Jr get in trouble, walking from one room to the next ( a bit longer of a trip than in a small house). Again none of that alone is terrible, but how much practice do you need walking in line?
And how many years (money spent each year) do you need to spend on basic math? Average kid in America spend 8 years, never even getting to algebra, I spent one, got all the concepts (after number recognition) done in a year, then I was free to move on. certainly saved my parents loot on repeat math programs

.