A more formal answer than what Jaroslaw Komlar perfectly describes.
Diamagnetism is a property of matter, superconductivity is a state of matter.
Diamagnetism describes how a material reacts when plunged into a magnetic field. Paramagnetism says almost the same thing. The difference between para- and dia-magnetism is just the sign of the response of the material with respect to the magnetic field. A diamagnetic is repelled by a magnetic field, whereas a paramagnetic is attracted by a magnetic field. The other way around : a diamagnetic material tends to repel the applied magnetic field, whereas a paramagnetic material tends to favour the applied magnetic field. The difference between dia- and para-magnetism is measured as the response of the magnetization $M=\chi H$ for a change of the magnetic field $H$, in terms of the (magnetic) susceptibility $\chi$. When $\chi < 0$, the material is said to be diamagnetic, when $\chi>0$, the material is said to be paramagnetic.
Clearly different now, superconductivity is the state characterised by a perfect diamagnetism below a given temperature (at some normal pressure condition, for weak magnetic field, ...)
You could have chosen as well to characterise the superconducting state with the voltage $V$ response to an applied current $I$, when the susceptibility is usually called the resistance: $V=RI$. For positive resistance, you have a Ohm metal, if you imagine that negative resistance could exist (only non-linearities allow for fictitious negative resistance to exist, so we already outside he scope of the linear response theory as discussed here) you may say you have an amplifier. Superconductor is the state characterised by a zero resistance.