There are a couple of arguments that might be presented to explain how affirmative action is different to apartheid.
1) affirmative action does not remove any rights of those groups not targeted by affirmative action. It is possible to argue that by supporting one group, you are implicitly disadvantaging the out-group, but usually this distinction is important. E.g. having a "ladies night" at a bar is generally more acceptable than a "men pay double price" night.
2) affirmative action policies tend to target education and employment. The right to vote is typically is given higher priority when it comes to balancing different rights.
3) Affirmative action can be seen as targeting equal outcomes, not equal treatment[1]. My country offers free breast cancer screening to all women above a certain age. Men can certainly catch breast cancer, but it's far more dangerous to women, so a gender-blind outcome of "we want less people to die of breast cancer" has a gender-aware policy implementation. Most people wouldn't say that this policy is "medical apartheid" on the basis of gender. Similarly, you could say that affirmative action is about getting disparate groups to have equal outcomes in terms of whatever metric you care you measure (educational attainment, income, etc)
[1] not everyone agrees with this. Some people argue that preferential treatment should be given to groups that were historically persecuted (e.g. land confiscation), as remediation.
