If anyone cares Ic an give you background on what 141 certification means in the aviation industry. I have been a business owner in the same field, just helicopters instead of planes, for damn near two decades.
The main thing to understand is flight schools teach under either part 61 of the FAA regulations of part 141 of the FAA regulations.
Teaching from part 61 is straight you student has to meet the requirements as set forth in the book of federal aviation regs. You'll here people refer to this as the regs is Far-AIM a lot in the industry. This is what most instruction is done under. It requires no approval by the FAA for a school to teach under part 61.
In a part 141 school, the facility will present a syllabus they create to the FAA for approval. If that is approved through the lengthy process 141 status will be granted.
Both options will get you the same result if you are looking for your pilot's rating. Part 61 however opens you school up to accept federal money like the VA bill as seen in the email someone posted previously.
I usually get a couple emails or calls a month by veterans looking to use their GI/VA bill for training and as said we operate in the much smaller helicopter field, and we also are a part 61 operation though and unable to take the VA stuff.
Also of note is you will see a lot of 141 schools cater to foreign students too. A 141 school tends to more be a pilot factory versus being a flight operation where instruction is just a minimal part of the business.