beachwood pro's:higher end area(hotels,shopping,rest.ect)  
                         con's:futher from downtown(da!)and airport
p.s.not sure about l.e.
downtown pro:closest to all area's and can walk to shopping,house of blues,rest.
                con's:it's downtown,parking a bitch 8-5 weekdays

I've stayed in all 3 areas and I like Beachwood and Independence the best because they have the jacuzzi's in the rooms

As evidenced by these beautiful 'Hastily Made Cleveland Tourism Videos'.

http://www.youtube.com/watch?v=oZzgAjjuqZM
http://www.youtube.com/watch?v=ysmLA5TqbIY
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/sarcasm