Both short descriptions have been edited for clarity from www.domino-printing.com
Data Matrix – Can store a maximum of 2,335 alphanumeric characters
A Data Matrix is a 2D barcode that encodes data in black and white, or contrasting dark and light, cells arranged in a grid. Unlike 1D barcodes, Data Matrix codes are omnidirectional, meaning that they can be read from any angle. The number of rows and columns within a Data Matrix increases with the amount of information stored in the code; this is limited to 2,335 alphanumeric characters.
Data Matrix codes are usually square, or sometimes rectangular in shape, and are made up of multiple light and dark square dots, or ‘cells’, arranged in a grid or ‘matrix’. Data Matrix codes are defined by an L-shaped pattern at one side of the code, creating two solid adjacent borders. This ‘finder pattern’ helps camera scanners to locate the code.
QR Code – Can store a maximum of 4,296 alphanumeric characters
A QR code, or Quick Response code, is matrix-based 2D barcode that encodes data in black and white, or contrasting dark and light, cells arranged in a grid. Like other 2D codes, QR codes are omnidirectional meaning that they can be read from any angle.
Though often printed in black and white, QR codes and can be printed in several different colors. However, there must be sufficient contrast between dark and light cells to accentuate the code itself and the background so the code is easily picked up by the scanner.
As you can visually see, there is a big difference between Data Matrix and QR codes so when you are referring to Gucci, Chloe, Michael Kors and most luxury branded handbag tags, these are all Data Matrix.