What is the difference between a transformer and a coupled inductor?
The two are basically the same class of device, although each will have parameters optimized differently.  The two names are to explain the different intended usage, which also gives you a quick guess of how some of the parameters may differ.  Of course only the datasheets would tell you what the parameters are for sure.

A transformer is specifically intended for transferring power from one winding to another.  You want the coupling between windings to be as good as possible, the leakage inductance zero, and the absolute inductance of each winding with the other open is often not a large concern.

With coupled inductors, each winding is still used for its inductance alone, although of course some coupling is being utilized else there would be two separate inductors.  Generally leakage inductance is less of a issue.  In fact, it can be useful to have some minimum guaranteed individual (non-coupled, or leakage) inductance for each winding.  The absolute inductance of each winding with the other open is also a important parameter that will be well specified.