Better to Visit Taiwan in June vs. August

I know that Tawain in June is rainy season and August has higher temperatures and is in the middle of typhoon season.
1. But in general, is it better to visit in the later half of June or later half of August?
2. What about in terms of overall travel cost, which month is better?
3. Lastly, during the last half of June, what type of rain (long constant drizzel throughout the day, or short thunderstorms, etc.) does the area get?
Thank you.