They all mean "way" or "method", but each have their own nuances. This post is explaining the difference between 方法 and 方, so I guess we could focus on the difference of 仕方 with the other two. Are there any other words that could be similar and have a slight different meaning?
Since you have read the link you provided I assume you are comfortable with adding 方 to the masu-stem of a verb to mean 'way of doing verb' e.g. 食べ方 (way of eating). The question is, what happens when you have a する verb? You might think something like 勉強しかた would be correct but, just like other uses of する, this one is irregular. The correct way to write it would be 勉強の仕方. So, quite simply, 仕方 is used with する verbs.
You could replace 仕方 with 方法 to get 勉強の方法 but this is just the formality difference discussed in your linked post. Replacing 仕方 with 方 would be ungrammatical.
仕方 also appears in set phrases like 仕方がない ("it can't be helped").