How is trangender female different from normal female biologically?
Remi gives a good introduction to the various terms and definitions you should know when learning about gender identity. However, he didn't really address your basic question:


  What is biological difference between transgender and a normal male/female[?]


Except for some likely genetic differences, which we aren't even close to beginning to understand, there is no biological difference between two persons born with $XX$ sex chromosomes, one of whom identifies as female (cis-gender) and one of whom identifies as male, or somewhere on the spectrum between female and male (trans-gender). There is no difference physically - both have the same organs, including sex organs, and both have the same secondary sex characteristics. One cannot tell the difference upon gross physical examination, X-ray, blood test, etc. Gender is a social construction, and while the majority of people in the West have a gender identity that matches their biological sex, some do not, for reasons that are not yet clear.