Both By the way and Speaking of are informal, spoken markers — natural in everyday conversation and casual messages, but too chatty for formal writing (there you would say Regarding… or In addition…). By the way means roughly 'here is a fresh, extra thing I want to mention'; use it to add a new topic, often out of the blue and slightly off the point. Speaking of means 'on the subject we just mentioned…'; use it to introduce something connected to what was just said. The difference: By the way jumps to an unrelated topic, while Speaking of stays linked to the current one.
Bringing in a new topic
By the way — By the way, Amina moved the meeting to Magaff Library.
Speaking of — Speaking of buses, the route to Ghafwick Station changed.
Not ""Amina moved the meeting." (dropped in with no signal, so it sounds abrupt)" — say ""By the way, Amina moved the meeting."". By the way flags that you are switching to a new topic, so the change feels smooth, not random.
Try it: Lars: The concert tickets arrived today. Mei: ___ concerts, Hiroshi is performing tonight. Which reply adds a linked new point? (Speaking of / By the way / Anyway) Answer: Speaking of — 'Speaking of' introduces a new topic connected to what was just mentioned, moving from concerts in general to the concert tonight. 'By the way' adds something unrelated, but here the new point is linked. 'Anyway' moves back to the main point, the opposite of adding a new one.
"That reminds me"
That reminds me is an informal, spoken marker — friendly and conversational. It means 'talking about this just brought something back to me'. Use it when the current topic makes you suddenly remember a new thought or a task you need to do. It links the new topic to a memory, not just to the words that were said. (In formal writing you would drop the marker and simply state the point.)
That reminds me — That reminds me, I need to return a book to Magaff Library.
Not ""That reminds me of my old school." (this describes a similarity, not a topic change)" — say ""That reminds me, I need to print the tickets before Friday."". As a topic-changer, That reminds me brings up a new task you just remembered, not something that merely looks similar.
Try it: Yasmin: Let's meet on Friday to finalise the plan. Diego: ___, we should print the tickets before Friday. Which reply reacts to a sudden memory? (That reminds me / Speaking of / By the way) Answer: That reminds me — 'That reminds me' is exactly for when the topic triggers a memory of a task, moving from Friday to printing the tickets. 'Speaking of' needs the same word repeated to link the topics, which does not happen here. 'By the way' adds an unrelated aside, but Diego is reacting to a memory.
Getting back to the point
Not every topic change moves forward — sometimes you steer back. Anyway is an informal, spoken marker that means 'enough about that — back to what matters'. Use it to leave a finished side-topic and return to the main point of the conversation. It is casual and can sound slightly dismissive of the side-topic, so it belongs in speech, not formal writing (there you would say To return to the point…).
Anyway — Anyway, we still need to choose a meeting time.
Not ""Speaking of meeting times, we still need to choose one." (this opens a topic, it does not return to one)" — say ""Anyway, we still need to choose a meeting time."". Anyway closes a side-topic and steers back to the main business. It is a signal to refocus, not to branch off.
Try it: Wei: ...and the weather's supposed to be lovely all week. Amina: ___, let's return to the plan for Senville Park. Which reply steers back to the main point? (Anyway / By the way / That reminds me) Answer: Anyway — 'Anyway' drops a finished side-topic and steers back to the main point, exactly what Amina wants. 'By the way' would open yet another new topic, and 'That reminds me' would introduce a remembered task, so both move away from the plan instead of back to it.