它们指逐渐结束某一活动或状态，常含永久性终止之意，后接动名词或动词不定式均可，且意思基本不变。但是，硬要找它们的区别，那么下面说法比较流行：
cease to do 长时间，甚至永远停做某事。
cease doing 短时停止做某事，以后还会接着做。特别强调主观控制的结果。
请看经典例句：
When you cease to dream you cease to live.
当你停止梦想的时候，你就停止了生存。
She ceased to think, as anger transmuted into passion
当怒火转化为激情，她停止了思考。
On his retirement the post will cease to exist.
他退休后这个职位就会撤掉。
That department has ceased to exist forever.
那个部门已不复存在。
For all practical purposes the treaty has already ceased to exist
实际上协议已经不复存在。
Once our services cease to be useful to them, we're expendable
一旦我们的服务对他们不再有用，我们就可能被抛弃。
It hurt so badly he wished to cease to be.
他觉得疼痛难忍，恨不得死了算了。
The great empire has ceased to be.
那个伟大的帝国已不复存在了。
He really gave his all till his heart ceased to beat.
他真是鞠躬尽瘁，死而后已。
I was married to a man who had ceased to attract me.
我嫁给了一个我已不再钟情的男人。
She ceased to be a member of the association.
她不再是协会会员。《外研社英语同义词辨析词典》
You never cease to amaze me!
你总能让我感到惊奇！
It never ceases to amaze me what some people will do for money.
有些人为了钱什么都干得出来，这一直使我惊愕不已。
He ordered his men to cease fire (= stop shooting) .
他命令手下停止射击。
The secrecy about the President's condition had ceased to matter
总统的健康状况已经没有保密的必要。
The girls ceased chatting for a moment when their teacher passed by.
姑娘们在老师走过时，停了会聊天。
The company ceased trading in June.
这家公司已于六月停业。
A small number of firms have ceased trading.
一小部分公司已经停止贸易。
The old woman ceased breathing the night before.
老妇人前一天晚上停止了呼吸。《外研社英语同义词辨析词典》
这个例子说明其接不定式或动名词意义区别不是太大。