Yes, there are a few key differences when you link two issues vs. connecting a PR to an issue.
When you link two issues together, they will remain in the column where the "parent" issue is in. Even though you will use GitHub's Closing Language to connect two issues, one issue can't close another or move it to a different column. If you connect a PR to an issue, you can choose to have the issue close automatically when merged. (More info on this here)
When connecting a PR to an issue, you're able to use the Waffle Connect Keywords or GitHub's Closing Language. You can ONLY use GitHub Closing Language when connecting two issues together, the Connect Keywords are not recognized yet.