Mechanical

I would say that manufacturing is the actual making of a product - forming/assembling/finishing etc., while production is the entire process of acquiring the raw materials, making the product (as above), inspection/test and preparation for despatch.

Production also includes various feedback loops - if a fault is detected at the inspection stage for example, the cause must be identified and rectified - be it something wrong with the raw materials or something done wrongly during manufacture.

Others definitions are available.

Interesting, at a casual glance I'd have reversed your definitions.
Suffice it to say they are broadly synonymous, and were I writing a report I would feel able to apply my own distinction with impunity. They are subsets of eachother, but which way around is anyone's guess.
Del

broadly synonymous

Can I use that?

All production units are part of manufacturing sector of a country, Does it give some clarity?

The terms are sometimes interchangable, but not always. Manufacturing refers to the creating of a product, which may involve mixing of chemicals, fabrication of parts, assembly,etc. It is also correct to say you are producing that product. But if you are growing and harvesting a crop, or extracting oil or minerals, you are producing those products, not manufacturing them.