Updated: May 23, 2022
First off, let me start by saying how unfair of a question this is. "Better" is mostly subjective,
there are socially accepted standards that dictate if something is better than something else,
but the majority of it comes down to personal preference.
This question has made the rounds and most coffee websites like the Perfect Daily Grind have done a good job in breaking down what is a Pour Over, what is a French press, and how each one works so I won't get too specific, but it is important to understand that these are not necessarily types of coffee drinks but brewing methods.
A pour over is normally referred to something that you drip through with a kettle. The medium ground coffee lays on the filter which can be made of a variety of materials as well as different micron sizes.
A French press refers to a small, simple and cheap brewing device that works via immersion, where water and coffee steep together to produce a uniform extraction.
The French press is a very popular brewing device because it produces a tasty cup with oils and a full body, since the coffee is brewed in the water, all the small coffee solubles with these traits go right through the mesh filter and end up in your cup. On the other hand the small pores in paper filters mean less oils, fats, and other solubles make it the whole way, leaving you with a cleaner, more balanced cup.
The French press is great option for someone who enjoys a rich chocolaty cup of coffee, someone just starting their coffee journey and not wanting to spend a lot of money, or maybe someone that has not yet tasted the intricacies of a stellar cup of coffee we now have access to.
However, once you are ready to elevate your coffee experience, you can move onto a more precise brewing method, with a measured fusion brewing device such as a V60 or Chemex.
If you really prefer immersion, you can try the Clever brewer which both brews via immersion but uses a paper filter to avoid getting too many different types of particles in your cup and allow you to taste what your roaster intended to share with you.
So, is a pour over better than French press? Again, personal choice, neither method lesser than, each one great in their own right with different cup results. At the end of the day, it is all about the coffee and who it is drank with.