Word documents were never intended to do what you’re doing. They were never meant to distribute documents to others for reading.
Your client is on the right track: that’s exactly what PDF is for.
In a nutshell: it’s all about the printer.
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Different printer, different look
Word processors like Word are generally designed to produce documents to be printed. When Word displays a document in a print layout or page view, it uses the characteristics of the currently-selected printer to determine what the document will look like when printed.
Printer characteristics vary a lot. Default margins, paper size, and other differences in both capability and configuration can make a document appear very differently when viewed or printed on one system as compared to another.
Different system, different look
Another common difference is fonts, which are not the same across systems.
If you create a document using one font that happens to be installed on your computer, and then view it on another system where the font is not present, things will look different. Word will substitute something “close” to the font you wanted. Unfortunately, “close” is vague, and can be startlingly different from what you intended.
The solution: PDF
The PDF file format is specifically created to solve this problem. PDF, which stands for “Portable Document Format”, is designed to display exactly the same everywhere, even across different operating systems, no matter what your system or printer characteristics.
Current versions of Microsoft Word and other word processors can save to PDF format directly. PDF creation acts like a printer — but a printer that’s the same everywhere. The interface used to save as PDF often looks very much like an interface you use to print the document.
The resulting PDF file can be viewed anywhere with a PDF reader and should look, and even print, exactly the same as your original PDF.
What PDF is not
PDF is not a format designed to be edited. Depending on the document, it can be, to some limited extent, but that’s not its purpose at all. Consider it a display-only format — not unlike the paper it’s intended to replace.
If you do need to exchange a document such that others can edit and make changes to it, Word’s “.doc” and “.docx” formats are what you need; just don’t expect the document to look the same everywhere.
The bottom line
Use the right tool for the job.
When sharing finished documents with others, use PDF. Always. Author the document so the PDF comes out the way you want it to, and then share that with your client.
If you need to share editable documents in Word format, just realize they will not display or print exactly the same everywhere.