There is a machlokes Rishonim whether emunah in Hashem is a mitzvah or not. The Rambam and Ramban sayig that it is. The Bahag (a Gaon), R' Kreskas, Tashbetz and others saying that it's not or that it can't be. What nafkah minah comes out of this machlokes?
First of all, the question of which mitzvos make up the 613 quota might not have any practical ramification. The Rambam writes in his introduction that his motivation for setting down the 613 commandments is just a means to keep the Torah's laws organized. However, most commentaries printed on the Sefer Hamitzvos believe otherwise, so I'll answer the question specifically regarding "emunah".
The most important difference is due to a question asked by R. Crescas, R. Albo, Abarbanel, and so many others on the Rambam, who count "Anochi" as a mitzvah. They ask, how can one command someone to believe? First of all, it's useless, because either someone already believes in God, and if he doesn't, a 'mitzvah' to do so won't help. Additionally, it doesn't make sense to have a mitzvah for something that is outside of one's control; these rishonim thought that a person can't just will himself to believe that something is true if he already believes it to be false.
In order to address this question, some have explained that when the Rambam writes that the mitzvah is to 'believe', he is actually referring to something other than mere 'belief'. Thus, the Malbim (Shemos 20:2), Maharam Schik (Mitzvah 26), Abarbanel (Rosh Amanah ch. 17) and Seforno (beginning of Ohr Amim) all explain that the commandment of אנכי ה' אלוקיך, believing in God, is actually a command to justify those beliefs, either philosophically or otherwise. (The Rambam himself discusses this in Moreh Nevuchim, specifically 1:50 where he states that in order to fulfill the mitzvah one has to be absolutely certain that God's existence is true, even though one does not have to be able to prove it). If this is not be counted as a mitzvah, perhaps there would be no requirement to justify one's belief, and can be left simply as 'belief'.
Shut Shoel U'Meishiv (Tinyana 1:51, "l'maskil echad") takes the opposite position. He also asks if there's a practical difference between the Rambam and Behag in this regard, and concludes that while according to the Behag, one can come to belief in God through philosophical justification or the like, the Rambam holds that one must belief because it is a mitzvah to do so, and no other reason. (!)
Alternatively, because it's unreasonable to think that there's a command to simply 'believe that God exists', the commandment is to believe that the One God is He Who revealed Himself to our forefathers at Sinai and gave them the Torah. (See Semak Aseh 1, Sefer HaIkkarim 1:14, Drashos Haran no. 9). I would imagine, however, that this is not a nafka minah and Rav Crescas would imagine that someone would have to believe in this as well.
Another possibility raised by R. Elchanan Wasserman is that the mitzvah as counted by the Rambam is not referring to an intellectual belief so much as it is a command to be reasonable not to be 'bribed' by the temptations that would be associated with denying God's existence. (I find this difficult for many reasons, though, and I'd love for someone to explain it to me)
Another actual difference might be regarding non-Jews, or Noahites. If one assumes that Jews are commanded to believe in God as a separate commandment instead of the foundation for observance, than this requirement might apply differently to non-Jews. While few would say that non-Jews are allowed to be atheists, but if one were to differentiate between the laws of idolatry for Jews and non-Jews, that would be the reason (this is stated explicitly in Shu"t Shoel U'Meishiv T'lisaah 2:29).
This is really a more general question about minyan hamitzvos--what is the nafka minah ever? It is well-known that the Gra objected to the enterprise of minyan hamitzvos on similar grounds. Various acharonim propose various explanations (see: hakdamah of Kiryas Sefer of the Mabit, hakdamah of R. Yerucham Perlow to Sefer Hamitzvos le-Rasag) but I don't think they would help here. In the end of the day, it might just be to know what the Taryag mitzvos are.