Writing is any example of creating a text.
Drawing-up means creating a text intended to serve a clear purpose as well as possible, according to standards understood by a community.
The difference is one of formality and discipline, as is illustrated by the examples that you give: constitutions, company costumer [do you mean ‘consumer’?] policies, contracts, wills, business plans, action plans.
Each of those is a more or less formal document, with certain conventional requirements. Any of them can be said to have been written, in the sense that someone strung the words together. To say that such a document has been drawn up, however, is to stress the fact that it has consciously been constructed or formulated to some recognised standard.
‘Drawing-up’ is related to the term ‘drafting’, both terms deriving from draw, meaning pull, with the implication of making effort. When it comes to texts, anyone can simply write something. Drafting, however, is a matter of explicitly aiming for a durable result. A ‘first draft’ is so called because it is a first step in drawing-up something lastingly effective. It is common to hear that a first draft was sufficiently fit-for-purpose to be adopted without further changes: even a first draft is more formal in intent and purposeful than simply ‘writing’ something. In policy formation, a text that is adopted is identified as the final draft or the accepted draft. That does not mean that it is in any sense experimental, sloppy or incomplete—and it might be the first draft, unmodified.
I was once employed for a few years to create speeches and other documents for the CEO of a large institution. The term in my contract was drafting. Writing can be as casual and ephemeral as you like. Drafting or drawing-up is always work, with formal intentions.
These days I often write (or write-up) recent progress on some research project or other. That just means that I am recording it. When I start preparing it for publication, I consciously shift to drafting.
If you propose to spend today writing a business plan, then you are saying that you will work on the document, fairly unspecifically. There is nothing wrong with that.
If you say that you will spend today drawing up a business plan, however, you are accentuating the fact that you will, for example, include all of the appropriate sections, in a recognised sequence, and perhaps using a particular corporate style. By the time you are done drawing it up, this document will have reached a certain standard of presentation, and will be fit for a certain purpose. An informed practitioner will immediately understand the import and interrelations of the document's various components.
This might well also be true of your writing a document, of course, but drawing up a plan accentuates this consciousness and satisfaction of formal requirements. The same goes for the other examples you have given. Something functionally amounting to a will might exist as a set of notes, but drawing up a will means conforming to a recognised format and standard.
There might be a comparison with, say, writing a story (which could mean the whole thing, or as little as just noting a few possible plot points) or scripting a story (which strongly implies the disciplined development of a usable document for performance, in recognisable form).
I am not sure what you mean by ‘a robber plan/plot’. If you mean the plan for a robbery, Herman Lamm really does seem to have elevated the previously haphazard occupation of bank robbery very much to one of drawing-up formal plans in an established, tested and effective format. Where earlier robbers might turn up fairly opportunistically, Lamm essentially created a business process of analysis and formal documentation, at the centre of the robbery-planning process.