If I submit a sitemap.xml which does not contain all of the pages of my site, will this affect my search ranking?
For example: If my sitemap only contained pages that had been created in the last month and didn't include all the pages from the past year, would Google treat these missing pages differently?
What we need to remember, Google is very picky, and even if you submitted all subpages of your website Google will index only part of it at first anyway.
What is more important is to make sure all pages 'cross-reference' each other, so make sure you make good use of main navigation, breadcrumbs, footer links etc. Because this is the main way googlebot will crawl your page.
Personally I had situation where after updating sitemap.xml Google immediately said new link in my sitemap is already indexed (because googlebot indexed it on its own before it was placed in the sitemap.xml).
Conclusion - add as many links you can and don't worry about missing ones. Just make sure you update your sitemap.xml once in a while (or do it programmatically).
There will most likely be a difference, but it's not certain that the pages excluded from the site map will get a lower status than without a site map.
The Google bot limits the time that it spends indexing a site. It will start with the most obvious pages and work it's way down, until it reaches a set limit were it will stop looking.
By adding a site map you will make it easier for the bot to find the most relevant pages. When the bot is done with those, it will continue with the rest of the pages. Possibly it will have more time left for the rest of the site if the site map made the first part of the work more efficient.