The numbers 14.04 and 16.04 refer to the release (or version) of Ubuntu that you'll be deploying when you use that specific one-click installer. Both versions are LTS, or Long-Term Support releases, though 14.04 will reach it's EOL, or End of Life, soon since 16.04 was released and is now the most up to date LTS release.
How does that affect you? Generally, an LTS release is no longer supported once it reaches EOL, so unless you're tied to 14.04 for one reason or another, it's best to use the latest release (i.e. 16.04). By no longer supported, often that means updates and upgrades targeting that release and it's specific needs will no longer be provided by Ubuntu (not specifically DigitalOcean).
As far as requirements go, one difference could be that 14.04 is a 32bit release while 16.04 is the 64bit release. Why does that matter? 64bit releases tend to require more CPU/RAM by design and they can more effectively use it over their 32bit counterparts.