First, 遊び is actually a noun, not a verb. It is formed from the verbal root. There is a group of nouns in Japanese which is derived from the root and uses this form (continuative form) as a standalone noun. It seems more common in the godan verb category. Some common examples are:
休み (a rest) from 休む (rest)
違い (a difference, a discrepancy) from 違う (differ)
育ち (upbringing, growth) from 育つ (grow)
狙い (an aim) from 狙う (aim)
泳ぎ (swimming) from 泳ぐ (swim)
As for the difference between plain forms and masu forms, masu forms are usually used in situations which require a degree of politeness or formality. They are typically used between people who do not know each either well or in situations which require formality. This is not an exhaustive list of when to use that form.
Explaining the spectrum of uses for the plain form would require a very in-depth answer, but in general, it is used in casual forms of Japanese, in relative clauses, in modifying phrases, with nominalizers (as in your example), among others.
Finally, as to the question of the difference between using 遊び and 遊ぶこと, this is actually quite a philosophical question. In language, when do you use a verbal phrase instead of noun phrase if both are possible? One answer is that it is somewhat of a stylistic difference. You could argue that there may be semantic differences between the two, but it's not always easy to distinguish. For example, in English, would you say "playing video games is fun" or "gaming is fun"? Are they different? Why choose one over the other? Likewise, someone might prefer to use 遊ぶこと if they want to emphasize the action of the verb, or vice versa if they want to stress the 'thing' that is 遊び. Having said that, they are not interchangeable because of how they interact syntactically with other phrases. For example, the sentence 計算に違いがある is acceptable but 計算に違うことがある is unnatural (when intending the meaning "there is a discrepancy in the calculation"). So, in my opinion, the choice between 遊ぶことor 遊び is based on either stylistic or syntactic considerations, possibly on semantic considerations if there are clear differences between the phrases.