It seems to be that the new trend with websites is that there is a single "page" with multiple points that you can link to rather than the traditional multiple page format. Is there a reason to do one over the other?

I think websites set up like that have information that can be shown in a small amount of area. Also you only have one page to maintain.

If you don't have a lot of content, a one page site might be better than multiple pages with very little on them.
Some single-page sites make use of horizontal scrolling areas to get more content onto the page, rather than having a 'mile-long' site.
You could also consider PHP includes to swap out main content blocks if you want to have more content available within one page.
I would think that if you have enough content to spread over multiple pages, it might be better for SEO, since you could have unique page titles, descriptions, <H1> tags, etc. for each page.

Think about what's the best way to present your content – what will your users need to be able to do on the site? And, of course, what will work best on small screens.