Is there a discernible advantage of using the Jetpack JSON API over the WP API? Which one provides the best resource utilization and will be easy on the server?
one of key reason not to choose Jetpack JSON API is that it requires to connect to wordpress.com for API calls.
some usefully comparison here. WP API vs Jetpack JSON API https://github.com/WP-API/WP-API.github.io/blob/master/misc/comparison.md
why not to choose Jetpack, it is well written. http://joshpress.net/why-the-wordpress-rest-api-must-be-in-core/
WP REST API will be merged into Wordpress core. https://make.wordpress.org/core/features-as-plugins/
so practically, WP REST API would be merged into Wordpress core in ver 4.6. it would be good to choose WP REST API for new development.
PS, WP REST API also has javascript client, which is not in Jetpack JSON API https://github.com/WP-API/client-js