There's a very slight difference in SOME very well recorded and engineered tracks, but you need well above average gear. As in the usual Sony, Bose, Apple, and Sennheiser (Veilheiser), and similar headphones won't make the difference you would expect for the money spent. And .flack only amounts to the track sounding a tiny bit more "large" or 'open". It's a borderline imperceptible difference and frankly not worth the effort unless you're into the technical fiddling and have the money to spend 4x as much on your setup. And avoid wireless unless your headphones and phone both support LDAC. You're better off buying an actually good, wired pair of headphones, than chasing .flac for regular listening purposes.
I say wired because practically all wireless headphones are not acoustically well designed by default. They're made to look a certain way to appeal to average consumers, which heavily compromises physical acoustics. Then to make up for bad acoustic design they use a horrifying amount of internal Digital Signal Processing to get them to sound alright. I'm not against PEQ (DSP) in general, but wireless headphones try to brute force sound quality with it and it never actually works. Technics very recently showed off the Technics EAH-A1000, which they claim was physically designed acoustically first like a traditional wired headphone, but I haven't heard them so I can't vouch for them. They're also $450 which makes me think you may as well find something else that's also wired.
BTW, any motherboard or device that has an ESS Saber DAC or similar level of performance is more than enough DAC. And you don't need a separate amp unless the one you have is making weird sounds, or if you can't get your audio loud enough. And if loudness is the problem you also have to make sure your headphone/speaker can handle the level of power the amp can output, or they'll break when you or someone else cranks the volume/gain recklessly.
Yeah, audio is full of lies and even the knowledgeable guys in it can be untrustworthy out of extreme personal bias and paid sponsorship. There's a reason audiophiles are still looked down on when a $3,000 setup sounds 90% as good as a $97,000 setup.
