What is the difference between French-pressed and Turkish coffee?
You list the main differences already (and yes they can be both made from the same bean or blend).

The name turkish coffee refers to the preparation method ... and the grind/granulation/coarseness is adjusted to the method.


Coarse ground coffee - Turkish preparation:

I come from a culture where Turkish coffee is brewed in almost every
home, so I always preferably go for it. And sometimes not having the
fine turkish-style coffee powder at hand (living abroad) I have made (read: tried to make)
"Turkish coffee" with a coarser ground coffee powder many times -
the taste is often quite alright, as you can compensate for the
surface (exchange area) with adding a bit more powder or letting it
sit a bit longer. However, as for this preparation you leave the
powder in the coffee (also while drinking) the coarser coffee does
not sink easily ... so you get it all between your teeth, if you
don't filter it out somehow (which is then not the traditional
Turkish coffee anymore).
Turkish-ground coffee - filter preparation / French press:

On the other hand using a French press, the fine Turkish-ground
coffee escapes through the mesh. This is not too bad, as the super fine coffee will sink and you will end up with coffee brewed in the carafe (like the "mud coffee" or kafe botz that they drink in Israel), with the unnecessary addition of making the French press dirty. Used to prepare filter (drip) coffee, the
finely ground coffee clogs the filter paper.  (being the adventurous
type I have experienced all this already, but wouldn't dare
recommending it)


In both cases you get coffee as a result after some hassle ... and even the desired taste might be reached ... but using the right coffee grind with the right method makes everything way easier.

The best is if you can grind your beans yourself (in Europe you can find public coffee grinders in many supermarkets and stores) ... then you can always achieve the desired coarseness for the right method of preparation (drip brew, french press, espresso, turkish,...).

Some additional notes:


You mention arabica - it is the most widely cultivated and
commercially used species of coffee (Coffea arabica) - so you will likely find it used in most preparation methods; the
other species that you sometimes get is called robusta (Coffea
canephora) and has a much stronger bitter flavor.
Not only the specieas, but also the roasting is very important for the flavor. So with the same bean you can get very different coffee (from a very strong dark roast to a very light roast). In some places I have seen turkish coffee prepared by quickly roasting the dry coffee powder in a cezve before boiling water was added, which gave it an extra toasty aroma.
In some places, turkish coffee will be made with the addition of spices (e.g. cardamom), which you can find also already in some ground Turkish/Middle Eastern/Iranian coffee blends. This will quite likely make a big difference in flavor when compared to "French" coffee (or to some drip-coffee blends that are sometimes made with the addition of hazelnut or vanilla flavors).