to contrive: To imagine s.th. and put it into practice. To find how to do a thing, esp. how to get over a difficulty.

He has ~d a new machine for spinning. You must ~ to get over this difficulty. He ~d means for his escape.


to conceive: To comprehend, to understand, to grasp; to think, to suppose.
This general verb hardly goes beyond the meaning of 'forming an idea or plan in the mind'.

I can't ~ why you allowed the child to travel alone. I ~ that there must be some difficulties.
How on earth a fellow could make such an ass of himself Soames could not ~. (Galsworthy)

from: Klein/Friedrich, Englische Synonymik (München: Max Hueber Verlag)

As a native speaker -

There are many subtle meanings of these to words depending on context....

Contrive is plan or bring about and is mostly used in a negative sense (evil plan etc..)

Concieve is to give birth to (an idea or make a baby) which is usually used in a positive sense (conception, concept). Dont confuse it with Perceive (though many do).

Examples: I conceived a clever plan to help them win. Our stragegy was contrived to get the Germany and Italians to argue.