Same round. Depending where you get it, one may be hotter loaded than the other. Almost all commercial loads of recent manufacture are marked "9mm Luger" or "9 Luger". Older loads were marked "9Para" or "9mm Para". If you get some of the older loaded stuff, circa ~1900-ish", it may or may not be loaded slightly hotter. The original Nazi Germany loading for the 9mm was in the neighborhood of 124 grs @ +/-1350 FPS, much hotter than todays standard loadings.
Either way it wont matter as long as the weapon it is fired from was made in the last 20 years or so. The only time it may have been a problem is in some older or vintage guns.
Dimensionally and for all practical purposes, they are the same.