The API (active pharmaceutical ingredient, the chemical that is the actual medicine) is the same but different manufacturers might vary the formulation (so the composition of the “rest” of the tablet eg colour, filler, casing).
This manufacturing process is tightly regulated and the new formulation should be equally effective as any other you have used in the past.
However, some patients find that the change of manufacturing process and formulation of their tablets effects how well their medicine works or the side effects that they get. (Colour of tablet has a surprising effect, yes, really, also size of tablet eg easier/harder to swallow, taste, how easy is the packaging to open, all sorts of things.)
If you notice your medicine seems not to be working as well as it used to or you get more/different side effects, your prescriber might be able to specify specific formulations or manufacturers, or make other changes to your prescription. Other patients (?most) are completely unaffected when the manufacturer of their medicine changes from one prescription to the next.
^^ the above is true across classes of medicine not only psych meds.