Neither is grammatically correct. If the subject is "effects" then the verb should be the plural conjugation. Also it's just "needed", not "needed be":
The effects of the missile were more powerful than needed.
The effects of the missile were more powerful than they needed to be.
Aside from that, "needed" is just short for "they needed to be", so there is no difference. It's just a difference in style.
In some cases it would be better to use the longer version if, for example, the verb tense is important, or there would be some other possible confusion. Some examples, using a different verb:
The crowds were larger than expected.
The crowds were larger than was expected (by someone).
The crowds were larger than the organizers expected them to be.
The crowds were larger than they were expected (by someone) to have been.
There is no significant difference between these four sentences, other than some specify who was expecting the result, or avoid doing this by using the passive tense.