How useful is it to use the latest Composer version to maintain my sites?
drupal-project has a composer version check built in that currently requires 1.0, so you're fine with 1.1 for now:

https://github.com/drupal-composer/drupal-project/blob/94404dcbed7fe9eb9ea8d95016b661ae6d4f9e3f/scripts/composer/ScriptHandler.php#L75

That said, more recent versions have performance improvements I think and sooner or later, there might be a problem that will force you to update, and then it will be easier if you already are using a custom more recent version. On the other side, your hosting provider might also update that version from time to time, you might want to check that with him. E.g. in case of a security issue.

But no, for now, you're not required to do this.