"suýt" carries a sense of urgent/excitement with it and tend to pair with an action/verb and means "almost". While "gần như" goes with adjective.
I almost won the lottery = "Mình suýt trúng xổ số" I almost ram into that car = "Mình suýt tông vào chiếc xe kia"
He is almost blind = "Anh ấy gần như mù" After eating half a pizza, she's almost full = "Sau khi ăn nữa cái pizza, cô ấy gần như no".
"suýt" describes something terrible/negative that almost happens. With "gần như", it's hard to generalize its various uses. One can use phrases like "gần như chắc chắn" (almost sure), or "gần như không hiểu gì" (almost not understand anything), etc.
Again--why is this not 'cái ly rượu' because we've had to use 'cái ly' before??