Does omitting a relative pronoun change the meaning?

One of my students asked me if omitting relative pronouns like "that" changes the meaning of the sentence.

Here’s the issue, in the books I have read mention that in spoken English (assuming this means less formal English) "that: is omitted when it can be omitted (according to the grammar rules) and still make sense. However, when we go to school and start writing essays we are told to omit "that" if it is not needed to be more concise. So this makes the removal of the relative pronouns when not necessary more formal. Though, I do know that academic papers have a word requirement and the ideal academic paper submitted is concise and follows this rule, but we all know that we do not live in an ideal world and academic papers may be padded by the author adding or not removing unnecessary relative pronouns.

In short, is it formal or informal to remove relative pronouns in both speech and writing? Why? Where is the source? Does anyone really know?

My research yielded little but an interesting on article on Garden-path sentences via Wikipedia.