EIB and EBRD have different mandates. EIB's responsibility is that of a policy bank, in that they provide lending support to initatives that advances the interests of the European Union in particular. EBRD's mandate is that to support market economies (i.e. free markets) rather than that of the European Union in particular.

For instance, if the government of say, France, wanted to build a bridge, EIB would be able to support it, but EBRD would not, as that is a public organization (the French government) building the bridge. On the other hand, if the government of France was supporting, say, Blackstone, a private investment firm, where Blackstone would build the bridge and collect tolls, EIB would be able to support it (advancing Europe) while EBRD would also be able to support it (advancing free markets).

A situation where EBRD would participate but EIB wouldn't would be something like... Blackstone buying a Kazakhstan state-owned enterprise. Here, EBRD is advancing free markets, but this doesn't really help advance European interests.

So the two banks do have overlapping mandates (even in projects abroad, EIB may finance them if they are using European components or European companies can benefit from it) but they are not exactly the same and in the context of the fall of the Soviet Union, when much of the Eastern European economy was controlled by state-owned enterprises, the need to adjust them to private ownership was important for integration into the world market.