Interesting question. To a native English speaker they are very different, and yet one dictionary defines a "sip" as a "small mouthful", so I can see why it might be confusing.
A "mouthful" is obviously a compound of mouth and full, but it doesn't necessarily mean that you fill your mouth to capacity.
A "mouthful" is defined as "a quantity of food or drink that fills or can be put in the mouth". Basically, if you can comfortably put it in your mouth, it is "a mouthful". This could be the normal amount of food you could fit on a fork or spoon or a comfortable bite of a finger-food.
A "sip" only really applies to liquids (drinks, or soup), and by definition is small amounts. We talk about sipping (as a verb) from the edge of a glass or spoon, because the action of sipping is also quite careful and controlled so as not to take in too much.
There are situations where "sip" may denote being very careful and deliberate to take smaller-than-usual amounts, perhaps when someone is ill or recovering from illness. However, some drinks are meant to sipped slowly (whiskey, for example), and so saying "he took a sip of whiskey" suggests the normal amount.
Likewise, "mouthful" can mean a normal amount of food, but it can be used to suggest someone has taken in too much, for example, "that's quite a mouthful of food you took there!" Also, we sometimes use "mouthful" to refer to the food already in somebody's mouth. In that context, the implication may be that there is too much food, for example "he tried to talk with a mouthful of food" suggests he had too much food in his mouth to be able to talk.