I love all the chant traditions. I couldn't say I liked one more than the other. Prayer is prayer.
The Russian and Greek chanting traditions are motivated by different philosophies (I won't address intermediate forms such as Romanian). Russian chant is choral and focuses on group cohesion. In contrast, Greek tradition has one or a few chanters chanting at a time, antiphonally, and the skill of the individual chanter is highly prized.
As others have pointed out, there are many variations within each tradition. Even within Greece, there are many specific styles of Byzantine chant which may sound very different, and some of them are more representative of tradition than others. Noteworthy local styles include those of the Patriarchal church, Thessaloniki, the islands, Athens, and Constantinople (the Patriarchal church has a different style from the other churches in the city). Of these, the most authentic are the old chanters of the Great Church of Christ, especially Iakovos Nafpliotis. These are the people whom everyone either imitates or pretends to imitate. Mount Athos tends to have the most heavily Western-influenced style.
Generally, performance groups such as the Greek Byzantine Choir are not good standards for Byzantine chant. Their style is very divergent from oral tradition. Capella Romana definitely has a heavy Western influence (which is NOT bad, but is not representative of authentic Byzantine chant). Vatopedi and Simonopetra Monasteries are also very innovative.
I don't know whether to laugh or cry when I hear people railing about how Byzantine chant is so much holier than Western church music, and their favorite chant "group" is something like Capella Romana. (Not talking about anyone in particular, but I hear this a lot in real life.) Most of these people balk when they hear the actual sound of Eastern chants.
In short, anything you can buy on Amazon is probably way left of center. Chant is for church, not CD players, so if you're imitating something you got on Amazon, that may be the wrong approach.
I hope I haven't made Podkarpatska's finger-scratching nightmare come true.