Adjustments are where the teacher helps with your mechanics by moving or adjusting a body part that may be out of alignment (for example, if you put your knee at risk by leaning it too far in during Warrior 2).
During an Assist, the teacher stays with you for a longer period of time, helping you move even more deeply into a pose. This can be an incredible way to expand beyond the stretch, twist or fold that you could do on your own.
Both are very helpful for giving you a visceral feel (beyond just verbal instruction) for what to think about next time you do the posture. If you ever feel uncomfortable, in pain, or prefer not to be touched, definitely let the teacher know. Conversly, if you like what s/he is doing, feel free to ask for more pressure!