An electromagnet is a magnet only when current is flowing through it. Materials which are used to make electromagnets have particular properties, specifically low retentivity and low coercivity. Retentivity is the property of a material which makes it retain a bit of the magnetic field once the magnetic intensity reduces to 0. While coercivity is the magnetic intensity required to erase that amount of magnetic field the material retained when the external magnetic intensity was made zero.
Permanent magenta on the other hand are usually made from materials which have a high retentivity and high coercivity. Basically a permanent magnet is made by placing a ferromagnetic material inside solenoid and applying a field. Because of the aforementioned characteristics of a permanent magnetic, it gets a net magnetisation when the external magnetic field is removed.
I'd like to suggest you to familiarise yourself with the italicised words for a better understanding of the definitions you've mentioned.
Basically both are due to current. But one (electromagnets) stops being magnetic when the external current producing the field is switched off. While the other (permanent magnets produced by magnetic induction) remains magnetic when the current stops because of a net magnetisation.