One thing that osu! doesn't have that Elite Beat Agents has is story and art.
In osu!, there are great-looking levels, but I only know two (Don't Stop Me Now and I Can Be Your Friend) that are story-based. All Elite Beat Agents levels have a story and charming art, so people play not just for the rhythm aspect, but for the (usually) wacky story that goes alongside it. That's one of the reasons Elite Beat Agents is remembered so fondly.
Of course, osu! is good at something Elite Beat Agents doesn't have, which is the competitive and cooperative community. This is the reason why they can't replace each other; Elite Beat Agents is more charming in the story and art department, while osu! thrives in the competitive scene.
Also, osu! will never replace every single good rhythm game. Copyright aside, I have no idea how they'll replace games like Rhythm Heaven, which changes its mechanics every minigame.