Let's start with あきる (akiru), which also deals with the feeling of getting tired of something. You might use akiru when a character simply loses interest due to repetition. Suppose a character, let’s call her Hana, has to eat the same plain rice balls for lunch every day. After a month, Hana might sigh and say, “もう、飽きた (mou, akita) — Ugh, I’m so over it.” Akiru suggests a general sense of weariness with something – a dish, a routine, a person – that’s just become dull.

うんざり (unzari), however, goes deeper. It expresses a much stronger feeling of being fed up, almost to the point of being disgusted by something, or by a situation. Consider Hana again. This time, she's working a temp job at a dead-end office where her boss is constantly making the same sexist jokes. If Hana were to grit her teeth and think, “もう、うんざりだ (mou, unzari da) — I’ve had it!” that would be a better choice. Unzari comes from facing something that isn’t just boring, but also frustrating, irritating, or even offensive. The key difference is the level of emotional intensity. You would use unzari when you're at the end of your rope, perhaps close to losing your temper or walking away.

Now, let's look at くたびれる (kutabireru). This word is about physical or mental exhaustion. Picture our friend Hana again: this time she is spending the night cramming for a difficult exam. As dawn breaks and she puts down her pen, she might say “ああ、くたびれた (aa, kutabireta) — Ah, I'm worn out.” This emotion is about being tired, and possibly a bit defeated. It is closer to the English feeling of being “beat” or “pooped.”

Returning to our scenarios, kutabireru wouldn't be the right choice for Hana's office situation. Certainly, the constant sexist jokes could mentally wear her down, but the root feeling is not one of tiredness -- it's anger and frustration. Nor would kutabireru be very appropriate for Hana's rice-ball problem. Hana might be tired of the same lunch, but not especially *because* of the lunch. The rice balls are merely boring.

In short: If something is just boring, akiru is your word. If something is exhausting, kutabireru. Use unzari when the thing in question is not just tedious, but also irritating, frustrating, even repugnant.