Greetings from a fellow Great Falls Native! I am forced to live in Hawaii now, but Great Falls is still home.
Well there is a lot more to part 141 vs. part 61. 141, as mentioned, governs flight schools while part 61 adresses flight training. Sounds similar, but there are some key differences, and part 61 goes beyond just learning how to fly and earning your certificate.
A school approved under 141 does business quite differently than a FBO/Instructor offering training under part 61. They must have a permanent office, maintain records for a certain amount of time, provide specific facilities for students, maintain their aircraft under more rigid requirements, and have a designated chief instructor. Additionally they must use a registered, standardized syllabus in training that details what will be taught when, and to what degree the student must master that skill before moving on to the next lesson. The chief instructor, or other designee, will give the student "stage checks" along the way to ensure he has learned what the instructor has said he learned. The reward for this rigidity is a (slightly) lower minimum hours requirement, and some schools have self examining authority. The 35 minimum hours for a PPL from a 141 school is very deceptive because in all reality you will need 50 - 60 hours before you are proficient enough to pass the checkride.
Training under part 61 is less predictable and more personal in the long run. There is a set list of standards any student must meet before the checkride, but the instructor can tailor the way these are taught to the individual student. For instance a 141 sylabus might specify the student solos before the first cross country flight (or vice versa). But (as in my case) if the weather along the cross country route sours you can change plans and go solo that day instead. The tailoring may result in a better overall training experience, or you instructor might just stink and not train you right or let you slip on a standard or two. Generally, if you go with a 61 program your cost per flight hour and instructor fees will be lower. Since you really won't finish PPL faster under 141 this is a better option financially. However all the advanced ratings should be quicker and cheaper to attain under 141.
When chosing which way to go myself, I came across a quote from a former ERAU exec saying basically, on a scale of 1-10 a part 141 student will come out a 6 or 7, every time, but a 61 student could be a 10, could be a 1! In the end though, the choice will come down to who meets your needs. Airlines generally want you to have a degree. A lot of colleges offer 141 schools in conjuction with their Av related degree programs, as well as a prestigious alma mater for you resume (not to mention job placement programs!). They also make it easy to combine flight costs with student loans.
One last note, a pitch I make in any flight training thread, JOIN AOPA!!!!!!! They will shove information your way from left and right that will improve you learning experience and keep you flying straight and level once you have your ticket. Good luck and happy flying!