Learn English – just in case VS. should

When using "in case" as in "take this umbrella, in case it rains", you acknowledge the possibility of the action (i.e. rain). You're not certain it will happen, but it's not unlikely. You'll often see "in case" in instructions, rules, etc. - for example, "in case of fire, stop, drop, and roll".

When using "just in case", on the other hand, you emphasise the unlikeliness of the action. If you're "taking this umbrella, just in case it rains", you don't really expect it to rain, but you're taking precautions. It's also less formal, it's not something you'll see on a warning sign or in a rulebook.

"Lest" is somewhat poetic and archaic, but I think it puts a bit more emphasis on the undesirability of the consequences than the actual likelihood of the event. Don't quote me on that, though.

If anyone calls, tell him I am busy.

If anyone should call, tell him I am busy.

Now what is the difference between this two?

The first one is a normal conditional sentence, while the second one suggests that the event of calling is unlikely.

Notes from Practical English Usage by Michael Swan -

Should anyone call, tell him I am busy.

We can re-write this sentence as the following -

If anyone should call, tell him I am busy.

Note in the previous case Should anyone call...., we don't use calls.

Please visit this link for more information and examples of should used at the beginning of the sentence with if dropped, as you quoted in your question.

N.B Pam Peters in Cambridge Guide to English Usage suggests that the usage of should in conditional sentence is becoming less often. Conditional uses of should with if included in sentence now sound rather lofty, but the inverted should at the beginning of the sentence with no if included in that clause is still a neat way of prefacing a condition.