Sweet Home Chicago is a good traditional blues tune however the tab is for the lyrics. The majority of the song is instrumental.
Playing a blues tune is more than playing tabs. The tabs are just the notes to the lyrics. Blues harp is accompaniment to the song. It harmonizes with the lyrics or plays melodies/riffs between the chorus or verses.
Learn to play tabs first of songs you know and like. Once you get comfortable to where the notes are on the harp, you can start playing around trying to match melodies of songs. Once you become accustomed to playing straight harp, you can start learning bends. Bends are what makes a good blues harp sound. Instead of playing in first position (the middle notes of the harp) you play in second position (the lower sounding notes). Using bends you can blend the notes easier and make sounds that staright playing can't do.
The harp sounds you hear in blues songs can't be reproduced by simply playing the harp in first or second or even third position. The good Chicago Blues sound is actually distorted by using a microphone and amplifier.