Both sentences are grammatically correct. "Factories are closing down" means that the factories are going to cease operation and stay closed forever, and maybe be torn down. "Factories are shutting down" also means that operations will cease, but the meaning is more temporary, like they are turning off all the machines because there is a problem with them that needs to be fixed or maybe the workers are on strike. So if you want to say that the factories are closing forever, use "closing down", and if you want to say that they are closed now but will open again after some problem is taken care of, use "shutting down".

shut down can be temporary.

"Factories are shutting down" is just a little bit more formal, but, in most situations, "close" and "shut" mean the same thing. You can shut the door, or close the door; close the drapes, shut the drapes, etc. There are a few exceptions, however. You never say "The store is shut" if a store isn't open for business, you always say "The store is closed", and I've rarely heard somebody say they are "Closing down the computer" instead of "Shutting down the computer"

They both can be used in this context. However, 'shut down' is also a computer term, but not 'close down'.

I think both are correct.