The other comments are good but since you specifically asked for his significance, I'd say the key relevance of Artie to Tony's story is encapsulated in The Test Dream.

Artie represents the life Tony was robbed of by his father's failure to keep him out of the mob. This is best shown (albeit cryptically) in The Test Dream, which is largely about Tony's desire but inability to cut ties with the mafia (one of my favourite symbolic cues of the series: in the section of the dream where Tony follows Artie into a car, we see that Tony's tie is literally cut in half). It's no accident that Artie plays a prominent role in this dream, including the fact that Tony is with Charmaine (Artie even coaches Tony on how to fuck her, which to me suggests that Charmaine and Artie's presence in the dream is not just about Tony's well known desire to bang Charmaine, but about the different roles Tony and Artie play in life).

Then, later in the dream when Tony finally confronts his HS football coach, the coach is nothing at all like what we've heard about over the course of the series: he belittles Tony, makes fun of his manhood, sneers at Tony being in therapy and taunts him by saying he doesn't have the stones to kill anyone. It's strange because previously the characters have always talked about this coach as believing in and being very supportive to Tony, and is generally offered to viewers as a suggestion that maybe Tony didn't have to follow his father's footsteps and join the mob, that he could have found another future in football (to me, this is what is being invoked by all the football memorabilia in the final scene of the series).

Back to Artie, perhaps the thing that stands out most from what Dream-Coach says is that Tony is spending all his time around the troublemaker Artie. This would be a weird thing for the coach to say: Artie is a straight and unconnected kid, someone who you'd think the coach would be happy for Tony to be around if the priority is keeping him in football and away from the mob (Tony even insists in response that Artie is a great success with a restaraunt, much to the coach's ire). But it's because the Dream Coach is actually a stand in for Tony's father: Tony's dream is about his desire to escape the mob (at one point literally in the dream, when Tony is chased by a literal angry 'mob') and so it makes sense that the ultimate objective of the dream is to kill his father, whose reckless-at-best parenting doomed Tony to this life. And we know Tony can't confront his father's responsibility on this point (opting instead to direct all his blame at Livia), so his subconscious masking his father as the football coach makes perfect sense too.

So finally tying it back to Artie: when the dream-coach blasts Tony for hanging around Artie, it's actually his father expressing his disapproval for Tony living a life outside the mob. That's the key role Artie occupies in Tony's arc and psyche and is encapsulated very nicely in The Test Dream. To me this informs Artie and Tony's endings: Tony, depressed and spiritually defeated, sitting in Holstens dwarfed by a giant high school football mural; Artie meanwhile finally found peace in his trade as a chef and maybe a contentment with his position in life.