They really mean about the same thing, except "downtown" implies more of a "business district" while "inner city" usually has a more negative connotation.

inner city = dense urban area. downtown = a specific place; typically,the city's financial (and usually geographical) center

"downtown" is an American expression meaning the heart of the city, where the buzz is, where the nightlife is, and usually where there are shops and businesses. "town/city centre" is the British equivalent, but this does not have the same resonance in music and song ;-) "inner city" refers to old, delapidated housing areas in the centre of cities, from where more affluent people have flown, leaving it to the poor, the transient and the imigrant. It has, probably unfairly, a very negative connotation, usually referring to "the problems of the inner city"