flatpak is basically linux's app store. flatpak is easier to install because it's just a button press and everything on flatpak is safe, and it handles updates for you, so try to use it as much as you can.

There are a few common ways to distribute software for Linux, which I'll try to explain while leaving out the more complicated parts:

• Appimage: The program & all of its dependencies are put into a special filesystem image file. (think special .iso or .zipfile) Works on any distro & does not require administrator privileges, but has a large file size. This is somewhat close to an .exe file.
• Statically compiled binary: The program is compiled in such a way, that the program file contains all dependencies. Unlike an appimage, the program file is not an archive, so it does not contain any files within itself. So, all the libraries (small program parts) are placed in a large program instead of being zipped up. Can usually be run without proper installation, like appimage & .exe files. This is also a bit like an .exe file.
• Package: The program & asset files are archived, usually compressed .tar archives, (linux equivalent of .zip/.7z/.rar) which are extracted (unzipped) during installation. Sometimes a small included script is run during install/update/removal. Usually, architecture (x86, ARM, risc-V, etc.) & dependency information is also included. Common package formats include: .deb (Debian based distributions like Ubuntu, Mint, etc.), .rpm (RHEL, OpenSUSE, Fedora & derivatives). These packages are usually somewhat distribution & version specific. For example, you might be able to install a Debian .deb on Ubuntu, but if you have incorrect versions of dependencies, it either cannot be installed (you get a warning/error) or the program won't work correctly or at all. Often, package managers like apt, dnf & pacman get these for you, so you don't need to think of them as files, but you can also install package files like .deb or .rpm.
• Flatpak: This one's a bit complicated. Something of a mix of all of the above. Works anywhere, but you need to have the application flatpak installed to run programs installed this way. Flatpak programs are installed for each user, meaning that no administrator privileges are required. They also support sandboxing, meaning that a program only gets access to files & functionalities it needs, like on smartphones, but that is optional and not all programs make use of it.
• Snap: Also complicated. For an average user, mostly like flatpak. Only guaranteed to work on Ubuntu, but might work elsewhere.

So, how you installed a program may change how it works a bit. For example, the versions of dependencies you have can change the program's behaviour. Also, some configuration can often be done when compiling a program, like specifying whether to use Qt or GTK for drawing windows, or disabling bluetooth support. Different packagers (people who make appimages, flatpaks and/or paclage files) may choose different options here.

Sometimes flatpak programs may use old versions of dependencies. Also, I'm not sure if this is the case with Firefox, but Chromium's (Google chrome & derivatives, like Brave) sandboxing (security things) conflict with flatpak's own, so some of Chromium's security features are disabled in favor of using flatpak's own ones.

If the flatpak version of Firefox caused issues, I'd recommend trying the native version (package manager) instead of one downloaded from the internet. You can either do this from the graphical app store by selecting something like native, dnf or rpm instead of flatpak, or the native package manager with sudo dnf install firefox for Nobara, I think. Unlike flatpaks, native programs are installed for all users & require you to type your password during installation.

If you use an appimage or manually downloaded .rpm file, you need to take care of updates manually, by downloading a newer version like you did during installation. I would strongly advise against this, unless necessary & you know exactly what you're doing.

I think this answered your question, but feel free to ask if anything was unclear or you have other questions. I'm a programmer & I've used Linux for a while, so I should be able to answer most questions.

Edit:
Sorry for the wall of text. I hope it wasn't too jargony.

TL;DR:
The wall of text has context & things that might br good to keep in mind, but I'd recommend removing the flatpak version & the Linux .exe equivalent you're using, then trying sudo dnf install firefox.

Edit 2:
Nvidia can sometimes cause problems on Linux, but if a different version of Firefox worked, it is very likely not the case here. Sometimes switching from Wayland to X11 or vice versa might help a bit, at least until the next driver update. Otherwise, I'd recommend Wayland, as it's more secure, actively developed, has fancy features X11 lacks & can be a bit more efficient.

Glad you found a way around the problem!