Question: Are there some differences in sentences 

> 
> *You've got to be strong*.
> 
> 
> 

and 

> 
> *You should be strong.*
> 
> 
> 

Are they the same?

Answer: 
There are things we *need* to do, and there are things we *should* do. Think of "needs" as requirements, while "shoulds" are recommendations. 

So, if you want to convey that it is absolutely essential to remain strong, use one of these:

> 
> We need to be strong.  
> 
>  We must be strong.  
> 
>  We've got to be strong. 
> 
> 
> 

On the other hand, if you are recommending that we remain strong, use one of these:

> 
> We should be strong.  
> 
>  We ought to be strong. 
> 
> 
> 

And if you want to express confidence that our strength will not falter:

> 
> We will remain strong.  
> 
>  We shall remain strong.
> 
> 
>
They're not the same.

> 
> You've got to be strong.
> 
> 
> 

This means that you *need* to be strong, that is that strength is a mandatory requirement for the situation.

This expression is often used in reference to emotional strength needed to weather a current or expected crisis, though of course it can refer to physical strength.

> 
> You should be strong.
> 
> 
> 

This could be interpreted in more than one way depending on the context. It could mean that strength is desirable but perhaps not necessary if you have other qualities that apply. For example you could win a fight using brute strength or by using superior speed to knock out your opponent before they have a chance to use their superior strength. In my experience this expression is much less likely to be applied to emotional strength: you might say "You've got to be strong to cope with a divorce", but you wouldn't say "You should be strong to cope with a divorce."