Local storage is VM-resident SSD (Solid State Drives) and Premium is network attached SSD. Local storage provides the lowest reliability among the storage options and is only temporary storage that leads to loss if the node is shutdown, panics, is stopped, or has a double bit memory error, etc.... Local storage is only intended for test and development environments where data loss is not a concern. Premium storage is persistent and provides the highest level of reliability among the storage options therefore it is the best choice for production environments or environments where data loss is a concern . It provides data persistence independent of a running vm and allows shutdown of a VM between sessions to reduce costs.
For additional information, please see: https://docs.microsoft.com/en-us/azure/storage/storage-introduction