How to remove strange space symbols in Word
That symbol is called a non-breaking space (NBSP), and is, in Unicode U+00A0 (which, as you correctly pointed out, is different from a regular space, which is Unicode U+0020).
The non-breaking space is used when you want there to be whitespace between two words, but you don't want Word (or any other text editor that supports them) to put a line break or line wrap there. For example, they're useful if you want to write the sentence "This computer supports Microsoft Windows." without a line break between "Microsoft" and "Windows".
As for your problem with seeing them in your Word document, it's very easy to fix. All you need to do is copy-paste and example of an NBSP into the search part of Find and Replace, and then type a regular space into the replace part. This should clear up your problem.
Some more information about NBSP, if you're curious, can be found at Wikipedia. However, it is useful to know for your purposes that while HTML treats any number of consecutive regular spaces as one space, it handles NBSP specially (you may have seen it represented in your HTML output as ). This is so that users can have a bit more fine-grained control over spacing in situations where CSS isn't suitable (or if you just want a quick and dirty hack :) ). So you may find that NBSPs do come in handy in HTML eventually - however, they definitely clutter up your HTML source, and they're annoying when not needed.
You may also like to know how to insert NBSPs in Word yourself - without having to copy-paste from Wikipedia all the time. You can insert them from the Insert Symbol dialog (Insert tab > Symbol > Special Characters tab > Nonbreaking space). You can also use the shortcut Ctrl+Shift+Space. In the Find and Replace dialog, you can also insert them by clicking More >>, then the Special dropdown, then Nonbreaking space.
If you have Word search for ordinary spaces, it will find nonbreaking spaces as well. So you can convert all nonbreaking spaces to regular spaces by simply putting a space in both the "Find what:" and "Replace with:" fields.