Although the concept is similar, we use these words in very different ways. A cruder distinction might be this: 'Ostensibly' is often something bland or positive which the people themselves are claiming about their actions. For example, the bosses are claiming that their trip to Hawaii is for business purposes. We suspect that this ISN'T true, but we have no way of proving it. 'Allegedly' works the other way round. It is usually used for negative allegations. We might say 'He allegedly blackmailed his employer'. In this case we suspect that it IS true, but it has not yet been proved. As Ben says, news reporters often use this term for legal reasons.

They are very similar in meaning! "Ostensibly" might mean "on the surface" or "according to their cover-story" - the word doesn't necessarily imply that anyone has said such a thing. "Allegedly" definitely means that someone has said it, or "alleged" it. Using "ostensibly" might imply that the real reasons are other than those stated. "Allegedly" is often used in public for legal reasons, when discussing crimes someone may have committed.