'Let' and 'Leave'

We use both let and leave in different ways and for different purposes. They cannot be used interchangeably.

Let + infinitive
A very common usage of let is in the phrase let us or let'swhen we are making a suggestion involving others.
We say this instead of Why don't we…? or I suggest we…which is quite formal. It is often used with shall we? as a question tag.
Compare the following:

• Let's just have a cold salad for supper this evening, shall we?
• And let's go for a run before we eat!
• OK. Let's do that!
• Let's forget I ever said that, shall we? I didn't mean to offend you.

When it is used with the negative there are two alternative versions to choose from: don't let's or let's not. Both are very common.

• Let's not get too involved in their argument. It's better if they sort it out themselves.
• Don't let's go to Sheila's party tonight. Let's just have a quiet evening at home

Let is also commonly used to make a suggestion to oneself in the phrase let me or to a third person in the phrase let him/her/them. Note also the usage with the infinitive of there is/there are.
Compare the following:

• Do you like this outfit?
• Let me see. I like the orange dress but not with that hat.
• I'm going to sell my car. Do you want to buy it?
• I'm not sure. Let me think about it.
• There's still a stain on this jumper.
• Let me try to get it out with this stain remover.
• Can Joey and Phoebe stay overnight next weekend? Oh, please let them stay.
• Let there be no doubts in your minds that we shall win this battle. 

Let = allow/permit
We can see from these last examples, particularly the Joey and Phoebe example, that let also means allow or permit. These are more formal alternatives and require to before the infinitive.
Compare the following:

• Let me say how pleased I am to see you here this evening.
• Allow me to say how pleased I am to see you here this evening.
• Permit me to say how pleased I am to see you here this evening.
• I wouldn't let them stay up after nine to watch the adult film on TV.
• I can't let you go to France without me.

Note that with the passive voice, we have to use permit or allow:

• We didn't let him go home until he had spoken to the Headteacher.
• He wasn't allowed/permitted to go home until he had spoken to the Headteacher.

Let me know/ Let me have
Finally, let is frequently used with know, where it means tell, and have, where it means send or give.
Compare the following:

• Please let us know as soon as possible whether you are able to accept our offer.
• If you had let me know earlier, I would have saved it for you.
• Can you let me have those reports by midday on Friday, please?
• Let me have half an hour to think about it and then I'll letyou know.

Leave = go (depart/quit/abandon)
As we saw with let, leave has a number of different meanings and uses.
Compare the following:

• The plane left early as everybody was on board half an hour before take-off. (= departed)
• Nobody leaves school at the age of sixteen now, like they used to. (= stops attending)
• Don't tell Maureen I'm leaving her. (= abandoning)

Left = remaining
Here it is almost opposite in meaning and is used as a past participle normally at the end of the clause, often with there is/are or have got:

• I haven't got any cash left. Can you get the sandwiches?
• There were only two days' rations left, but they had to last for six days.
• Nothing was left of the castle. It had been completely destroyed.

Leave = let it remain
It is here that the meaning of leave comes closest to let, close but not identical.
Compare the following:

• I'll eat later. Just leave it for me in the fridge.
• I left my car in the car park and took the bus into the town centre.
• I can't make the decision. I'll leave it for you / to you to decide what to do.
• I can't get the stove to work.
• Leave it with me / to me. I'll deal with it.

This final example combines a number of different usages of let and leave:

• Let me finish off the translation for you.
• OK. There are only four pages left. I'll leave it for you. I have to leave now anyway!

There are even more shades of meaning of leave than we have covered. Check them out in a good dictionary.