It sounds as though you are typing the string "FALSE" or "TRUE" into the
cell when you type it yourself. This is not the same as the Boolean
values False and True which you will get as the result of an IF or by
using the Boolean values  in a statement as you have done in

> if cell  1 = cell2 then TRUE else FALSE

If you really want the words TRUE or FALSE in the cell, then you need to
assign them as strings ...

> if cell 1 = cell2 then "TRUE" else "FALSE"