Here is a passage that offers an opportunity to understand the practical difference between an independent and subordinate clause. Writing clearly depends on balancing the relative importance of the thoughts we want to communicate, the independent clause representing the central point around which other thoughts in the sentence—and the thoughts in adjacent sentences—organize themselves:
The new owners of the restaurant were taking every opportunity to greet customers personally. Recently, one of them was high up on a ladder to change a light bulb in the ceiling while he shouted hello over his shoulder to a couple just barely in the door.
We have two problems to consider here: the construction of the second sentence, and the relationship of that construction to the first sentence. A clause, remember, is a group of words with a subject and predicate, the predicate being that portion that contains the verb for the subject, together with any other words necessary to complete the assertion. Every clause represents a thought. There are two kinds of clauses, independent and subordinate, and they are named such because an independent clause succeeds by itself in stating a complete thought, whereas a subordinate clause only supports the independent one in some way. We can recognize the difference between the two kinds of clauses by the conjunction, if any, with which the clause begins. In this passage, the word while is a subordinating conjunction, and so makes the clause it introduces of second importance to the main clause.
The independent clause of the second sentence tells us that one of the new owners was standing on a ladder, and the analysis of subject and predicate would look like this: one of them | was high up on a ladder. That should help us see that this fact, this assertion, was central in the writer’s mind. Not only is it the independent clause, but the writer has placed it as the first clause in the sentence, the one we read first and so put most emphasis on. The second clause, the subordinate one introduced by the conjunction while, conveys the thought that he | shouted, and we are to understand that this action is secondary to the fact that one of the owners was high up on a ladder. Why? Because compositionally the fact of shouting has been placed in a subordinate clause.
But is this the best way to organize the scene? To answer that, we need to look at the first sentence. The predicate of this preceding simple sentence is were taking every opportunity to greet customers, so the idea in our mind as we come to the period of that sentence has to do with greeting customers. The first clause of the next sentence, though, has not to do with greeting customers, but with being high up on a ladder, and so the flow of ideas has been interrupted. One principle of good composition is cohesion, the close connection of related ideas, and the writer could have more closely approached that ideal by changing the thought that constitutes the independent clause of the second sentence. If the second sentence is organized instead around the idea of shouting hello, its central point would now begin with a thought closely related to the one with which the previous sentence concludes: Recently, one of them shouted hello over his shoulder to a couple just barely in the door, while high up on a ladder to change a light bulb in the ceiling. Or, better, this trimmed up version: Recently, one of them shouted hello from a high ladder to greet a couple just barely in the door. Shouting, it seems, is a way of greeting, but in both revisions we read it first and move more smoothly between the two sentences.
Was the original version grammatically incorrect? Not at all. But grammar and style are closely related, the one affecting the other and neither always being preeminent. Good revising will consider them both, and make its changes with purpose, audience, and effect in mind. A complicated affair sometimes, but with complication often comes a richer subtlety.