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.
Technically they are the same thing it depends on its usage.
We typically think of an inductor as storing and releasing energy so for example in a typical switch mode fly-back type power supply we might call it a "fly-back transformer" or "coupled inductor" rather than a transformer.
Another example is the output inductor on a multi-output buck converter. If we decide to wind the inductors for different outputs on the same core we would call it a coupled inductor.
Whereas normally for a transformer we apply an ac voltage to the primary to generate one across the secondary and power transfer is instantaneous. Any energy it stores is usually considered a bad thing (causing losses) while inductors (coupled or otherwise) are intended to store and later release energy.
A coupled inductor stores energy. They typically have a gap, where the energy is stored in the magnetic field. Other than that, they do look very similar to transformers. A coupled inductor would be used, for example, in a flyback converter, where it stores energy while the switch is on, then dumps the energy to the output when the switch is off.
Most transformers (other than coupled inductors) are wound on low reluctance cores. They do have magnetizing and leakage inductances, but these are more like parasitic effects. An ideal transformer does not have these characteristics. An ideal transformer does not store energy.
On the other hand, a coupled inductor is an inductor, and is designed to store a significant amount of energy in the core flux. Because of this, the core has a gap, either a discrete gap or a distributed one, like in a powdered iron core. Energy is stored mostly in the gap.
I think most of us would regard a coupled inductor as a special type of transformer.