A power adaptor is a devise or means by which any utility can be used / accessed without any changes to the property of the utility, only used to match the devise. While in the power converter some of the metrological properties will be changed either type or quantity.
I would simply make out the difference as
Adapter - change of AC to DC power with required voltage level but specification like voltage and currents are fixed for example SMPS which is commonly found in solid state circuits. The power adapter is simply a connector that changes the plug shape to match the outlet. It does not change the voltage or electrical output in any way.
Converter - is also knows as rectifier (broadly) which has the controlled output with use of thyristors or SCR's or IGBT's etc hence controlled output. Converter is an electrical device that changes the form of an electric signal or power source, as by converting alternating current to direct current, or an analog signal to a digital signal.
Hence if I use any converter without a controlled firing circuits can be broadly set into adapters.
PS- converter can also be referred to inverter (DC to AC conversion) then above specification varies. Adaptor, convertor, rectifier, inverter, transformation and transducer, yes, there is confusions between them! However, if each one is properly defined, difference between them can be easily achieved.