The question everyone actually hasThe Certificate Is Identical
A private pilot certificate earned under Part 61 and one earned under Part 141 are the same certificate, with the same privileges and the same limitations. Nothing printed on it records which rule you trained under.
Both routes end at the same practical test, administered by the same designated pilot examiner, against the same Airman Certification Standards. Your logbook records dual received, solo, cross-country and night — not a regulation number. When you later apply for an instrument rating, a commercial certificate or a job, the application asks what you hold and how many hours you have, not which part of 14 CFR your first school operated under.
No airline distinguishes between the two on a resume. No insurer prices a renter or an owner differently because of it. No employer has ever asked a candidate to produce their training school's air agency certificate. We say this plainly because it is the anxiety underneath most of the searching people do on this subject, and the answer is genuinely reassuring.
The one place the two routes diverge in the long run is the restricted ATP, and it is worth being precise. FAR 61.159 requires 1,500 hours for an unrestricted ATP certificate. FAR 61.160 allows a restricted ATP at 1,250 hours for a graduate of an approved associate degree program and 1,000 hours for a graduate of an approved bachelor's program at an FAA-authorized institution of higher education. That is a university route, not simply any Part 141 school. Our students take the 1,500-hour path, and we would rather tell you that here than at hour 900. If that route matters to your plan, read the ATP 1,500-hour rule before you choose a school.