Actually, the answer is not obvious. @RoSiv gives the textbook case of symmetric cell division, where the two new cells can indeed be considered identical, and this is valid in many cases. But there are also cases of asymmetric cell division, where the "mother" and "daughter" cell are clearly different.
In asymmetric cell division, the parent cell is polarized already before division by gathering certain proteins at one end of the plasma membrane. Therefore, during replication one of the the two new cells obtains certain proteins while the other does not, which makes them distinct. See this article for details.
Asymmetric cell division occurs among other things during embryo development, where it is important to establish body plan asymmetry. It is also the basis for the stem cell concept: stem cells are thought to divide asymmetrically so that the "mother cell" remains a stem cell (stays immortal) while the "daughter cell" becomes mortal and differentiates.