The difference is typically expressed as the difference between "is a" and "has a". Inheritance, the "is a" relationship, is summed up nicely in the Liskov Substitution Principle. Aggregation, the "has a" relationship, is just that - it shows that the aggregating object has one of the aggregated objects.
Further distinctions exist as well - private inheritance in C++ indicates a "is implemented in terms of" relationship, which can also be modeled by the aggregation of (non-exposed) member objects as well.
