The difference is usually due to a time gap (or not).
TIME > - - - - - - - - - - - - - - - - - - - - - - - - - - > .......... [ I was sick.] ............................X You were only sick for a while. Then you were not sick. Then you said so.
...........[ I ... have .. been ... sick ....... ]X You were sick from the time you started to be sick until now, wh