My unfettered nerdiness made me answer:
It was an issue because elves only have eternal life on Arda (the world). They can be reborn under certain circumstances (if I remember correctly), but they are tied to the world, so when they die, it is a fairly deep tragedy. I would imagine that Arwen losing her immortality might also mean a true, complete and permanent death for her, but I'm not certain. Aragorn is human. They live shorter lives in the world, but they return to Eru (the supreme deity in LotR) when they die, and live there forever. So Arwen's sacrifice is a rather serious thing, and Aragorn might be the most selfish asshat ever for letting her go through with it (although I do appreciate that she actually makes an autonomous decision in this). He already gets his eternal life after his body dies. Though it makes for a rather pleasantly poignant love story. I only wish that Arwen got to have an actual personality (beyond being soooooo pretty
) to go with it.
Been a long time since I read these books, though. I may be misremembering things.
However, that does make me wonder how they will handle the immortality issue with Solas and Lavellan, assuming that there is even a scenario where they will be able to continue their relationship.
... what? I'm totes on topic. Shut up. 
While most of this is correct I just wanted to clarify that Elrond was considered one of the half elven (peredhil). Only Elrond, his parents, his brother Elros and their descendants would get the choice to be judged with the fate of the eldar or fate of men. This was due to Elrond's father and his part with one of the silmarils; not even other half elves would get this choice. Therefore when Arwen made her choice to die she chose the fate of men, she would leave the circles of the world with Aragorn and the death of her soul would not be permanent. However Elrond and his wife etc. would never see their daughter again, not even in death at the ending of the world. Arwen and Aragorn will always be together at least.
You are correct though, the real tragedy was the separate afterlife of elves and men. That is why when Galadriel's brother (a full elf) Aegnor falls in love with a mortal women (Andreth) and leaves her (much like Solas leaves Lavellen) he can never be with her, even after they both ended up dying ....now that's depressing. He even refuses to be reborn and instead chooses to stay in the halls of Mandos forever because he does not wish to live in a world without her.
EDIT: I would also like to add that although it is a belief of the elves that they might cease to exist after the ending of Arda, they are not certain. I would like to believe that Eru is not the kind of ass that would damn them to such a fate anyway.
Yeah sorry about that people but when I think I find another Tolkien fan I tend to get excited. Excuse mistakes, on my phone at work <.<