'Tint' suggests a delicate change in colour, perhaps to make the natural colours appear warmer or brighter. 'Dye' is a more radical change in colour.
In the past, people used to tint photographs to add colour or to deepen the colour. This is different from dying, when a product - such wool or cotton - is saturated in a colour to change it from white to blue, for example.
If you tint your hair, you maybe add a slight reddish tinge to natural brown hair or gold highlights to blonde hair. If you dye your hair, you change the colour completely.