An immersion (or locally closed immersion) of schemes is a morphism that can be factored as , where is a closed immersion and is an open immersion. If it is moreover an open morphism, it need not be an open immersion:
Example. Let be a nonreduced scheme, and let be the reduction. This is a closed immersion, whose underlying set is the entire space. Thus, it is a homeomorphism, hence an open morphism. It is not an open immersion, for that would force it to be an isomorphism.
Remark. However, every closed immersion is a closed immersion; see Tag 01IQ.
Maybe you should change your title.
The title is exactly as intended.