apropos

The OED defines apropos as: to the point or purpose; having direct reference to the matter in hand; pertinent, opportune. It comes from F fw à propos, to purpose, plan. Purpose comes from L rw prōpōnĕre, to set forth, propose. The combined etymologies would read as: to that which was proposed, set forth. The word has been in English since lC17.
Apropos is the direct link between two occurrences or ideas; it is pertinent and apposite. It is more than simply on topic, it is entirely relevant to the subject of the matter. Usually, it reinforces a point made in an argument, by providing an example or filling a previous hole in the argument. When something is apropos of another, they are strengthened in their similarities in reference to their context. In the relevance to the greater idea, this is the ideal connection. On the other end of the spectrum, the word can be used to interject something more vaguely related. This sense is used quite loosely, more often in casual conversation than literary uses. Here the word is used laxly, with more of a by the way intonation.
Another important factor of something that is apropos is the spontaneity of the connection. It is not simply a branch off of the main topic, or a digression that once bore some resemblance to its original idea, the connection is established after both ideas are set in motion. There is no preconception of the connection. It is not planned, but effectively incorporated into the plan. However, it can be used to pick up a proposed idea in conversation to segue into a discussion about it. 
Similar and relevant are more common words, usually used instead of apropos, but are not quite as strong. Similar means that two things bear some resemblance, but the similarities may not necessarily be relevant to the topic. Relevant is slightly stronger of a word than similar, but one thing that is relevant to another still not quite as precise and pertinent as something that is apropos of another. 
Apropos most often implies precision — one fact or idea directly related to the topic that arrives at an opportune moment. It is a connection between two things that strengthens both and the idea behind them being discussed. It is to the point and doesn’t detract or digress. It can segue or relate one topic to another, but the underlying messages and intentions are preserved by its direct relevance. 

Citations:

“apropos, adv., adj., and n. : Oxford English Dictionary,” http://www.oed.com/view/Entry/9957?