Resumes are sales tools. Look at other sales tools you're familiar with. Imagine you're at a conference, and a salesperson at a booth hands you a business card. You'd expect some level of formatting, in order to help convey a level of professionalism at least, if not some overall brand message for their company.

What would you think of the salesperson if they gave you a business card that just had plain unformatted text plopped on it?

Now, unless you're applying for jobs doing page layout, no one is going to be highly critical of your resume's formatting and layout. But, there is a level of expectation among recruiters and hiring managers, that you've put thought and effort into how you present yourself during the hiring process. This is important because it's taken as a sign of the thought and effort you'll put into doing your job, should they hire you. Hence, it's worth presenting at least a minimally-formatted resume.

If you're worried about continuity, it's easy to choose a word processor with a history of stability and backwards compatibility (ie MS Word). Worst case, you end up having to re-build your resume from a pdf or - gasp - a printed copy.

All that said, there may be some merit in keeping a plain text version of your resume, as well. Many job applications require you to enter details of past jobs into an online web job application, and cutting/pasting from a word processor sometimes results in relics that are painful to edit back out (ie a bullet list may collapse and look messy). Having a plain text version can make that easier.

Which brings up a good point - it's best to not think of your resume as a single (static) document, but rather as a collection of information that you need to present with slight tweaks for different audiences. For instance, I have a "baseline" resume with all of my employers/positions/projects with detailed notes on each. It's probably about 5 or 6 pages long. When applying for a job, I run through it and remove content that's not relevant (a process that might take 10 minutes), allowing me to create a document that's focused on the specific opportunity I'm going after. The end result is a typically-long resume that's focused on a specific position.