The audio quality from the dedicated audio output and the headphone output should be pretty much exactly the same. The dedicated output will be located on the back of your TV, as opposed to the front or the side like the headphone socket. It's maybe not the case for yourself, but I'd much prefer to have any audio cables connected & hidden around the back of the TV as opposed to sticking out the side/front.
Also, a TV's speakers are usually disabled when anything is connected to it's headphone output. If you connected speakers to the headphone socket and were planning to revert to the built-in TV speakers at all (say for quiet TV viewing) then you'd have to unplug the speakers each time.
I had a look at the Logitech website but couldn't find the exact form of connections available on the speakers. Normally multimedia speakers like these come with a standard male 3.5mm jack for connecting directly to a PC's sound card. If this is the case, and you bought a standard 3.5mm (male) to 2x phono (also male) cable - then you'll need a separate stereo coupler to connect your 2 male 3.5mm jacks. If the Logitech's happen to have a standard 3.5mm input (female) as well, then the cable you bought should connect directly from the TV to the speakers without any other form of adaptor.
I have a different set of (different) Logitech speakers and in addition to the standard 3.5mm jack, they were also supplied with a short phono to 3.5mm jack convertor cable. Maybe you got something like this as well with yours?
