The two words are used in pretty similar situations but have a very slightly different meaning. Adequate refers to something being just barely enough. Whereas sufficient means you have plenty and feel comfortable without worries.

If something is adequate, there is enough of it, but only just enough. If there is sufficient quantity of something, this suggests that there is as much of it as you need. Usage of these adjectives often means quantity rather than quality, whereas appropriate and suitable suggest a qualitative response to something.

Sufficient is a synonym of adequate. As adjectives the difference between adequate and sufficient is that adequate is equal to some requirement; proportionate, or correspondent; fully sufficient; as, powers adequate to a great work; an adequate definition lawfully and physically sufficient while sufficient is equal to the end proposed; adequate to wants; enough; ample; competent; as.