I believe the key distinction you want is the 'finiteness' of the expression. Complete sentences in English generally require a finite verb. According to some (cf. Klein 1998), the finiteness itself is what grants a sentence the power of assertion. Consider:
We ate the pizza.
The above is a sentence, and it has a finite verb, ate. Meanwhile:
Us eating the pizza.
This expression seems like it might be a sentence at first -- it has a subject, a verb, and an object. However the verb, eating, is nonfinite, and hence the expression as a whole only refers to an event. It does not state or assert that the event happened, as a sentence would. (Although stating/asserting are not the only options available to a sentence, of course.) Instead, we can use its reference within another sentence:
I remember us eating the pizza.
Here, our finite verb, remember, takes on the whole expression us eating the pizza as its object.
So considering this, a single noun, like towel, is generally unable to form a sentence. (In certain contexts, it might be fine on its own. However these generally involve an implied unsaid verb.) "No towel" is a more complicated situation, as no could have several interpretations based on the function of towel. However it is still apparent that the expression is not a sentence.
Then how do express that a towel exists? (or doesn't?) We invoke an existential clause. In English, this involves adjoining a form of the copula (to be). In our case we will use the singular present-tense, is. This is finite, so we are well on our way to sentencehood, but we also must observe that English finite verbs require subjects. Thus we use the expletive pronoun, there. It doesn't have a meaning on its own; it is merely there to satisfy the requirements of a well-formed sentence. So we now have the power to declare...
There is no towel.
People have argued about what makes a sentence for a long time. I just chose a particular argument that I thought had some weight. Some people would claim that finiteness is irrelevant and there are other key properties that make a sentence valid. (Some rely on verbs, some don't. There are plenty of languages where overt verbs aren't necessary to make a sentence.) Others would say that the definition of sentence doesn't matter - there are reasons you might say "no towel" even in situations where there was nothing else implied, so we shouldn't draw an arbitrary distinction of what a "sentence" is. Others still would say that your internal judgement of grammar isn't even as strong as you think, and it is really more like a rough guideline.
To answer your question: there are no concise terms to describe an issue so complicated. My answer is far from adequate for such a topic, but I hope it put you on a path to figuring out what you wanted to know. You also may be interested in reading up on speech acts, grammatical mood, discourse analysis... (The list could go on endlessly. )