Both is correct and has the samemeaning. However "こと" can be used for vast situation, so I think your adviser advised you the more proper way to express the same meaning.
Actually "話すのを止める" is more smoothing Japanese expression I feel.
It depends on the person, but I feel the following.
"抗議するのを止める" is more smoohter than "抗議することを止める"
"学校に通うのを止める" is more smoohter than "学校に通うことを止める"
"こと" also can be used for the following,
"日本語を勉強することを始める(止める)" = "日本語を勉強し始める"
"実行可能な計画を策定することが重要だ" = "実行可能な計画策定が重要だ"
We can make Japanese sentences without "こと", and at the same time we can make Japanese sentences with "こと."
Japanese sentences without "こと" sounds smooth for me (It depends on the person.)
Nowasays, more and more Japanese natives use "部分" like the following,
"安全に対する配慮が不足していた部分もありますので、今後はしっかり対応したいと思います"
Some people are sensitive for the exessive use of "こと" and "部分", but most of Japanese don't care abut it.
"こと"や"部分"が多用される日本語は、適切ではないことだと思いますが、それが今の日本で受け入れられている部分も認めなければなりません。(^^)