In general, semantics relates to what sentences mean, and pragmatics to how they are used. There is no clear boundary line as to where one starts and the other ends, because typically an utterance must be understood by reference to who is uttering it, to whom, on what occasion, in front of what audience, and with what common knowledge. Those who emphasise semantics tend to think of sentences themselves as having a meaning independently of their use, while at the other extreme there are some who hold that one can only speak of the use of a sentence, not its meaning. When Russell was writing, in his earlier years, theories of pragmatics did not yet exist, so he is taken as defending a strict semantic understanding of sentences. Pragmatics came along in the 1950s and 60s with John Austin's book "How to do Things with Words" and John Searle's "Speech Acts", and by Paul Grice's theory of implicatures and the cooperative principle. It is now a standard part of linguistics.
One of the powerful features of Grice's theory is that it serves to explain how utterances can be misleading even when they are true. For example, suppose Alice says, "Bob drove home and had a beer". We understand this to mean something different from, "Bob had a beer and drove home". In propositional logic, these two sentences are just "P & Q" and "Q & P" and so by the commutativity of & they should mean the same. We could try to adjust the semantics and say there is something different about 'and' in English that conveys the sense 'and then', but according to Grice there is no need. The fact that we understand the events as happening in the order stated is because the cooperative principle includes the maxim "be orderly". We are entitled to expect Alice to state the events in an orderly manner and it would be misleading, though not actually false, if she gave the wrong order. We can recognise the difference between the meaning of an utterance and an implicature, because an implicature is cancellable. Alice could say, "Bob drove home and had a beer, but not necessarily in that order". This would be expressly cancelling the suggestion of orderliness. If the semantic view were correct, i.e. that 'and' in English means 'and then', then such a cancellation would be a self contradiction.
Another example. Suppose Alice asks Bob, "Where is Charlie?" Bob knows that Charlie is in the library but he answers, "Charlie is either in the library or the kitchen". P entails P or Q (in classical logic at least) so Bob has not said anything false. But we understand his answer to be defective in some way. He should not say 'either/or' unless he does not know which. Again, we could try to bend the semantics and say that 'or' means 'or ... and I don't know which', but Grice comes to the rescue again and says that Bob's response is true but misleading. Another of the maxims of the cooperative principle is "be as informative as required". Bob is being less informative than he could reasonably be in the circumstances. Alice is entitled to suppose that Bob is following the cooperative principle and that he doesn't know which room Charlie is in. As before, meanings can be distinguished from implicatures by the fact that the latter are cancellable. Bob could say, "Charlie is either in the library or the kitchen; I know which, but I'm not telling". If not knowing were part of the meaning of 'or' then this would be a self contradiction.