Factory = where physical / tangible products are made, e.g. car factory, cake factory, clothes factory, chocolate factory (😊). Plant = where a process is carried out on something (e.g. water treatment), or where something intangible is generated (e.g. power). I think there is some crossover of terms. E.g. car plants (although I personally would say “car factory”!). As an aside, “plant” can also mean the heavy machinery you might find in a factory. This might not be the best explanation! Hopefully it helps a little.

There is definitely overlap and both words would probably be ok in your case. A 'plant' is more general. Common types of plants are assembly plants, production plants, manufacturing plants etc. A factory is a type of plant, and it would be the preferred word for the case where there is only one type of item being produced in high numbers (e.g. a ball bearing factory or a tire factory). You couldn't use 'factory' if there were no physical product, or if the work were essentially chemical rather than mechanical. 'Factory' seems to fit your case. On the other hand, if there is engineering, design, and testing onsite, a more general word could be better.

David and Don's explanations are good. The word "plant" is mostly used for talking about power plants, water plants, or recycling plants -- offhand, I can't think of any other places that are typically called "plants." We say "factory" when we want to refer to a place where products are manufactured. A place that manufactures security systems would be a factory.

Factories are where parts of a product are assembled to create the product such as a Car Factory where the parts of the car arrive from multiple sources and are then assembled at the factory to create the end product which is the finished car. Plants on the other hand are where the "product" is created through a process or a science if you want to call it so, for example a Power Plant creates energy through a process that eventually turns into electricity. In conclusion, it's very simple: Factories assemble products Plants create products Hope this makes sense to you now.