Atheism and anti-theism so often occur together at the same time and in the same person that it's understandable if many people fail to realize that they aren't the same. Making note of the difference is important, however, because not every atheist is anti-theistic and even those who are, aren't anti-theistic all the time. Atheism is simply the absence of belief in gods; anti-theism is a conscious and deliberate opposition to theism. Many atheists are also anti-theists, but not all.
When defined broadly as simply the absence of belief in gods, atheism covers territory that isn't quite compatible with anti-theism. People who are indifferent to the existence of alleged gods are atheists because they don't believe in the existence of any gods, but at the same time this indifference prevents them from being anti-theists as well. To a degree, this describes many if not most atheists because there are plenty of alleged gods they simply don't care about and, therefore, also don't care enough to attack belief in such gods. Atheistic indifference towards not only theism but also religion is relatively common and would probably be standard if religious theists weren't so active in proselytizing and expecting privileges for themselves, their beliefs, and their institutions.
When defined narrowly as denying the existence of gods, the compatibility between atheism and anti-theism may appear more likely. If a person cares enough to deny that gods exist, then perhaps they care enough to attack belief in gods as well — but not always. Lots of people will deny that elves or fairies exist, but how many of these same people also attack belief in such creatures? If we want to limit ourselves to just religious contexts, we can say the same about angels: there are far more people who reject angels than who reject gods, but how many nonbelievers in angels attack the belief in angels? How many a-angel-ists are also anti-angel-ists? Of course, we also don't have people proselytizing on behalf of elves, fairies, or angels very much and we certainly don't have believers arguing that they and their beliefs should be privileged very much. It's thus only to be expected that most of those who deny the existence of such beings are also relatively indifferent towards those who do believe.
Anti-theism requires more than either merely disbelieving in gods or even denying the existence of gods. Anti-theism requires a couple of specific and additional beliefs: first, that theism is harmful to the believer, harmful to society, harmful to politics, harmful, to culture, etc.; second, that theism can and should be countered in order to reduce the harm it causes. If a person believes these things, then they will likely be an anti-theist who works against theism by arguing that it be abandoned, promoting alternatives, or perhaps even supporting measures to suppress it.
It's worth noting here that, however, unlikely it may be in practice, it's possible in theory for a theist to be an anti-theist. This may sound bizarre at first, but remember that some people have argued in favor of promoting false beliefs if they are socially useful. Religious theism itself has been just such a belief, with some people arguing that because religious theism promotes morality and order it should be encouraged regardless whether it is true or not. Utility is placed above truth-value.
It also happens occasionally that people make the same argument in reverse: that even though something is true, believing it is harmful or dangerous and should be discouraged. The government does this all the time with things it would rather people not know about. In theory, it's possible for someone to believe (or even know) that a god exists, but also believe that theism is harmful in some manner — for example, by causing people to fail to take responsibility for their own actions or by encouraging immoral behavior. In such a situation, the theist would also be an anti-theist.
Although such a situation is incredibly unlikely to occur, it serves the purpose of underscoring the difference between atheism and anti-theism. Disbelief in gods doesn't automatically lead to opposition to theism any more than opposition to theism needs to be based on disbelief in gods. This also helps tell us why differentiating between them is important: rational atheism cannot be based on anti-theism and rational anti-theism cannot be based on atheism. If a person wishes to be a rational atheist, they must do so on the basis of something other than simply thinking theism is harmful; if a person wishes to be a rational anti-theist, they must find a basis other than simply not believing that theism if true or reasonable.
Rational atheism may be based on many things: lack of evidence from theists, arguments which prove that god-concepts are self contradictory, the existence of evil in the world, etc. Rational atheism cannot, however, be based solely on the idea that theism is harmful because even something that's harmful may be true. Not everything that's true about the universe is good for us, though. Rational anti-theism may be based on a belief in one of many possible harms which theism could do; it cannot, however, be based solely on the idea that theism is false. Not all false beliefs are necessarily harmful and even those that are aren't necessarily worth fighting.