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Both of these words are most commonly used as nouns. They refer to unpleasant sensations that you feel in your body, although they are slightly different feelings. A pain is usually used to refer to a kind of sharp discomfort that is difficult to ignore. For example:

Yesterday I suddenly felt a lot of pain in my stomach. I was taken to hospital where they discovered I had appendicitis.

Ache is similar to pain, but it usually is used to refer to a duller kind of discomfort that may continue for longer than a pain might. Common examples are:

headache, stomach ache, backache and heartache

Note that heartache is used to refer to a kind of emotional pain, for example if you are in love with someone and they are not treating you well you can say:

He is causing me a lot of heartache.

Less commonly, both pain and ache can be used as verbs. Pain in this form means to cause someone unhappiness. Note that you cannot use pain in the progressive or continuous form in standard UK or US English.

It pains me to think of you being so unhappy with your life – meaning it makes me feel unhappy that you are so sad.

You can also use ache as a verb. Unlike pain, ache can be used in the progressive form. For example:

My back is really aching.
I wish my leg would stop hurting, it really aches.

Interestingly, you can also use the expression aches and pains to refer to a general feeling of being unwell. Now let’s have a look at sick and ill.

SICK and ILL

Both of these words are most commonly used as adjectives, as in:

I feel sick.

In American English, these two words have more or less the same meaning – they refer to a feeling of being unwell, whatever the problem might be. On the other hand, in British English, they have more distinct meanings. Sick most commonly refers to feeling as if you are going to vomit, as in:

I think I’m going to be sick.

…while ill refers to any other feeling of being unwell. However, there are some phrases which are commonly used where sick has a more general meaning. For example:

I’ve been off sick for ten days – meaning I haven’t been to work/school for ten days because I’ve been feeling unwell.

Sick can also be used as a noun to refer to vomit. For example:

If you have children, you can be sure that you will have to clear up some sick at least once during their childhood.

HURT

Hurt is a little bit different to the others in that it is commonly used as a verb and an adjective and, less commonly, as a noun. Of the above words, hurt is probably most similar to pain and ache and it can also mean the same as injure or injured. The key difference between hurt and pain and ache is that usually if you are hurt, it means the discomfort you experience is clearly caused by something outside of yourself. However, pain and ache suggest a more internal reason for the unpleasant physical feeling. Here are two examples:

Be careful on that ladder, you might hurt yourself if you fall – here it is used as a verb

He was badly hurt in the train crash – here it is used as an adjective

When it’s used as a noun, hurt usually refers to some emotional pain. For example:

When he told her he wanted a divorce she could hear a lot of hurt in his voice.

As you can see, the five words we have discussed here are similar in their general meaning but are quite different in their specific use. I hope it' s bit clearer for you now and that my explanations haven’t hurt your head!