The quickest way to see that nirvana is not heaven is to notice that Buddhism already has heavens and does not aim at them.
Buddhist cosmology is populated with deva realms, celestial, long-lived, pleasant, and attained by merit. Everything that a reader expects the word heaven to mean is there and is on offer. Rebirth in one is a good outcome and the tradition says so.
It is still not the goal, and the reason is structural: the realms are impermanent. A being reborn in one remains within samsara, exhausts the merit that placed it there, and falls out. A heavenly rebirth postpones the problem in the most agreeable available manner. It does not solve it.
A tradition that has celestial reward available, describes it in detail, and then declines to aim at it is not offering its highest term as a version of one.
Nor is it annihilation
The other standing misreading, and the older one, nineteenth-century European writers regularly described Buddhism as a cult of nothingness.
The metaphor in the word is a flame going out for want of fuel. What is extinguished is craving, aversion and delusion, and with them the process producing rebirth. It is not the destruction of a self, because the analysis that produces the doctrine has already concluded there was no permanent self there to destroy.
When asked directly whether one who has attained the goal exists after death, does not exist, both or neither, the Buddha is reported to have set the question aside as wrongly framed, and to have asked in reply in which direction a fire has gone when it goes out. The question presupposes an entity whose fate is in question.
Where the traditions genuinely differ
Not over heaven, but over whether nirvana is anywhere else at all.
The Theravada account distinguishes nirvana attained at awakening, with life continuing, from the final passing at death. Madhyamaka in the Mahayana makes a much stronger claim: there is no difference whatever between nirvana and samsara, since neither has inherent existence, and liberation is therefore not departure to another condition but the correct seeing of this one.
That is a real disagreement between living traditions, and it is a more interesting one than the question the heaven comparison was asking.