Short answer, each platform requires you to buy the base game and expansions.

It's possible to play with just the base game (currently FFXIV: A Realm Reborn; after Stormblood Launch this will be called something like FFXIV Starter Edition). However, if you do not register Heavensward, you cannot enter Heavensward areas or use any HW-only features. Additionally, if you log out in a HW area, you can only log back in on a platform that has HW access. If you would like full access on both platforms right now, you need to buy and register both ARR and HW.

However, the Stormblood expansion will subsume Heavensward, so beginning in late June, each platform will require just (1) Starter (ARR) and (2) Stormblood (both Stormblood and Heavensward expansions as one). So if you don't need full access right this second, you can hold off on Heavensward because it will be included in Stormblood.