毎(まい)～の様(よう)にAlmost every ~, Nearly every ~, On a ~ basis

The main difference between these two grammar points depends on the context they are used in. While both can be used to express a sequence or regularity of actions, the former phrase "まい～のように" implies a high frequency or regularity such as "almost every day" or "not a day passes without," making it feel more intense and exaggerated. In contrast, "ごとに" translates to "each" or "every" and provides a feeling of a strict, fixed pattern or cycle, and can be used in situations when actions are done systematically or in a certain order.

Both "まい～のように" and "たびに" indicate a recurring situation. Nevertheless, when "まい～のように" is used, it suggests that the event or situation is repeating at a very high frequency, akin to 'almost/nearly every time' or 'on a specific basis'. On the other hand, "たびに" conveys the idea of an event or situation happening 'each time' or 'whenever' a certain condition is met, which may not necessarily have the same frequency or regularity implied by "まい～のように".

The phrase "まい～のように" implies frequency and indicates something that happens or is done nearly every day, week, month, year etc, denoting high frequency and regularity. On the other hand, "おきに" describes intervals or regular spacing between things or events, as in something happening every other day or every 2 days, 3 days and so on, highlighting the repetitive pattern and the space in between events or actions.

The phrase "まい～のように" is used to indicate frequency or regularity of an action, implying "almost every time" or "on a regular basis". For example, in a phrase like "週末まい日本へ行く" it means going to Japan almost every weekend. "につき", on the other hand, is used to indicate a reason or cause, like "due to" or "on account of", or to signify "each" or "every". For instance, in "暑さにつき、試合が延期された", it implies the game was postponed due to the heat, whereas in "一人につき一冊", it means one book per person.

The Japanese grammar point まい～のように is usually used to describe something happening nearly every single time or with a high frequency, for example, "I go to the gym nearly every day". On the other hand, ように・ような is generally used to make comparisons or analogies between things or situations. These could be actual similarities or desired similarities, such as, "She sings like an angel". Essentially, まい～のように deals more with frequency or regularity, while ように・ような deals with resemblance or similarity.