"Rank is a way so you know how you are against the rest of the osu! community. A lot of people use this (and level) to compete. Also, when used in multi, it can help you get people around your skill level.
For me, the rank that you have means nothing more than a silly competition. Yes, I want to grind and get my rank up so I'm at least in the top 5k for all the modes, but that's just my competitive side.
The rank on the songs on the other hand (for me at least) shows a more concrete way of showing you how well you are in the osu! community. After a while, you should be able to get in the top 1k in the ranks and (eventually) top 50. After you start to get into the top 50's consistently (country then globally), good job for you. You are now getting to be a decent player in whatever game mode you picked." ~DragonSlayer96