I'll start by defining conservative. A very succinct definition comes from Michael Oakeshott from his influential essay "On being Conservative".
To be conservative, then, is to prefer the familiar to the unknown, to
prefer the tried to the untried, fact to mystery, the actual to the
possible, the limited to the unbounded, the near to the distant, the
sufficient to the superabundant, the convenient to the perfect,
present laughter to utopian bliss.
So a conservative likes the status quo. He doesn't think the status quo is perfect, but he doesn't want to risk it unnecessarily in search of something better. He wants to conserve it. This definition, like all definitions of conservative is not clear and tight. How do you define necessarily? How much sacrifice is worth it to conserve what currently is? Won't that sacrifice in itself destroy the sufficient, convenient society? Society is constantly changing despite anything you do, how do you make reforms to bring things back to the known good? Unfortunately, there are not easy answers. Edmund Burke, arguably the father of modern conservative thought, says that a conservative must rely on prudence to navigate these difficult questions.
Prudence is not only the first in rank of the virtues political and
moral, but she is the director, the regulator, the standard of them
all.
So a conservative must prudently respond to attempts to risk his society in pursuit of utopian dreams. In Burke's time this involved incredible sacrifice in life and property to oppose the radicalism of the French Revolutionary states.
A reactionary, on the other hand, is a much simpler ideal. A reactionary merely wants to return society to a previous state of existence. This pursuit can, however, be in itself a utopian idea. Imagine trying to retun a contemporary European state to an absolute monarchy. Such a goal would tear the current societal order apart and reshape it into a new one distant in time and structure the status quo. The means for doing this could also be radical and would probably have to be to have a chance of success. Reactionaries need not prefer present laughter to utopian bliss, in fact they often are rather radical.
Conservative reforms, however, are often reactionary. Reactionary reforms are a return to something that has already been tried and known, conservatives prefer that to changing society along abstract concepts like liberty or equality.
Let's apply this to your example of abortion in the United States. Since this is a touchy subject, note that I'm trying to describe an existing opinion, not endorse it as correct. A conservative reactionary would attempt to prudently remove Roe v. Wade and, over time with lawful legal reform and return to a previous state of the country where the humans in utero were protected by law. After thinking about this problem, the conservative decides that even after 50 years society would not be upended by this change if the end was pursued with care. A radical reactionary might try to stack the court, secede part of the country from the whole, or seize power in an attempt to preserve the lives of the unborn. The line between their methods is not obvious and it relies on prudent statesmanship to distinguish conservative reform from mere reactionary action.
Note that in this case, there is also a very conservative position to leave Roe v. Wade alone. A conservative, even one who thinks abortion is immoral, might be wary of overturning a strong legal precedent and brand even the conservative reactionary above as rather radical. This conservative might think the prudent course is to change the culture of the United States with prayer or private education. This would promote reform with less risk to the sufficiently good, familiar order of society than the risk created by a contentious legal action.
