Gender, as a social construction, does not have a biological basis. Gender identity can move over the female / male limits assigned at birth. This does not necessarily imply a correlation with sexual orientation.
Transgender: someone who does not subscribe to their gender in terms of feminine or masculine. In transgenderism there is an adaptation from gender to personal identification, it can be combined, opposite to the gender assigned at birth, or any other combination. It is not a closed category.
Transexual: a person who finds that their sexual identity is inconflict with their sexual anatomy.