for the better（固定短语）意思是“好转,向好的方向发展；变好”，“change for the better”意思是：为了更好而改变。可以理解为：变好；向好的方面变化。
例如:
1. I am happy to say that the patient has taken a turn for the better, last evening it was touch and go whether he would survive the night.
我高兴地说，这个病人已经好转。昨天晚上情况十分危急，能否再活过一夜都很难说。
2. The situation is now so bad that any change is likely to be a change for the better.
目前情况恶劣, 只要有变化就是向好处变.
3. The situation is now so bad that any change is probable to be a change for the better.
现在的形势太糟糕了，任何变化都可能是往好处变。
4. I trust that things will take a turn for the better in the near future.
我期待着在不久的将来事情会有所转机。
The doctor felt that having a few days' rest would be for the better．
医生认为休息几天较好。
The world situation is changing for the better．
世界形势日趋好转。
His state of health has changed for the better recently．
近来他的健康状况好转了。