Criticize is a verb. Critique is a noun.

Critizing: "Your work is terrible". Critique: "While your work is not the best, if you practice doing this, this and this, you will improve in the areas you are lacking".

Criticise would describing the negative properties of something/someone - it might not be a fair comment. Critique would normally be a fair expert review of something. Viewers of a talent show might criticise someone but the judges would give a critique.

To criticise suggests a negative view. A critique is more like a review, it can be positive or negative.

Very interesting question! Critique is a detailed, analytical and balanced review. It is often used for literary, political and philosophical theories. It is also used for business presentations.