I went to Oundle 6 years ago and have met stoics since.
Both schools will produce a very different student at 18 with a slightly different path.
Oundle is MUCH more academic. It's not filled with pure academic, head-in-book brainboxes like Westminster (they do thrive there too), but due to, imo, the school's strongest asset and USP, which is an incredible work-ethic and approach to their work, sport, art etc. It is why people choose Oundle. It is the school with the strongest work ethic in the midlands I feel, and the consistently good grades are testament to this. Many parents are pleasantly surprised at how well their children do when they send them to Oundle. This is an intangible asset that most Oundelians take with them and subsequently probably why they also do well at University too. Other schools with good grades at a-level have students that struggle at university because the work ethic does not come from them but the school. At Oundle, the school engenders this into students. Oundelians move on increasingly across the pond and this doesn't look set to change, so if that is of interest to you, they will cater for it
Stowe, although sometimes bashed irl for being unacademic, does get good grades for its students but having spoken to friends from there, they were allowed to relax if they wanted to maybe a little bit too much and probably could have done better. However, this is anecdotal.
Stowe is a good school and beautiful, but most are bored by 18 and cant wait to leave as it is in the middle of nowhere. Oundle, being part of a town, staves off this boredom, especially with the ability to go into pubs (AT 18!), waitrose, tearooms. I was climbing the walls to leave at 18, however.
Both are Full boarding or day, no weekly or flexi anything and same set-up (i.e day students have own house not mixed into boarding houses). This, in my opinion, is preferable and helped to engender house spirit and streamline pastoral provision. Both are run as boarding schools and day students walk to the beat of a boarding school drum.
Oundle do not throw out students in an unfair fashion. When I was there every single person who was asked to leave, imo, was done so fairly. If your child doesn't get the grades and doesn't work hard, then they will be asked to leave, but if they showed the right attitude to their studies but were having a bad exam day, then the school contextualise this. Some parents would argue their DC worked hard and it was unfair but they are 2 hours away in a car and not privy to the day-to-day goings on and it seemed fair to me whenever a student left. Everyone is made clear at the school they expect you to work hard.
Art-wise I'm sure Stowe is wonderful but not sure it is going to produce another Damien Hirst. Ditto Oundle, although the focus is on Fine-Art. Stoics do always seem to follow their passions which is great and same with Oundle. Stowe definitely helps students do what they like which is always important.
Wider student profile at Oundle, more diverse range of interests in students. Very wealthy but also very understated. Stowe more hooray-henry but also pretty similar parents. Oundelians a little more cosmopolitan and a little more international.
Oundle does not cater for everyone so if the above doesn't sound right for your DC then I would send to Stowe.
AMAZING facilities at Oundle, I'm sure at Stowe too but don't choose a school based off this. Good luck with the decision. As long as your child is happy that is the most important thing and they normally know within 2 hours of going to a place if they click with the people and the energy. :)