If you have old Types 2.x relationships and ran the migration wizard (there were actually 2 such wizards, in Types 3.0 and again in 3.4) then the database structure is the same as for newly created relationships, so there is no difference between 1 and 2.
Details of the relationships are stored in the table wp_toolset_relationships, while the connections between posts themselves are stored in a combination of wp_toolset_connected_elements and wp_toolset_associations.
The post relationship field is just another way of creating a one-to-many relationship field, where the differences are just in the UI about how you enter the data (the simpler UI is only shown on the child post with a dropdown to select a parent). The data is stored in just the same way as "true" one-to-many relationships, in the same tables.
Meaning that functionally, in terms of performance or how to retrieve and display related content, there are no differences.
