"Manage" means that you have the control and "succeed in surviving or in attaining one's aims, esp. against heavy odds; 

"cope" is just means "deal effectively with something difficult"

The usage depends on the situation and context. 

You will have to cope with daily life if you don't have a good education.

Thanks to the medication, I was able to manage my anxiety.

Without medication, I have to cope with my bouts of anxiety.

To manage/ To cope with

I guess "to manage" is used in more wide senses than "to cope with'. But in common cases such as "to manage anxiety" or "to cope with anxiety" - which word is more convenient?
What is significant difference between them generally?

To manage/ To cope with

I guess "to manage" is used in a wider sense more wide senses than "to cope with'. But in common cases such as "to manage anxiety" or "to cope with anxiety" - which word is more convenient?
What is the significant difference between them generally?

***

They are very similar and there are probably situations where either one is fine. Intuitively I would say to "cope with anxiety". In my mind, the general difference is you would manage something you can control, and cope with something you can't control. 

Examples:

I would manage...

- my anxiety medication treatments

- my schedule

- the people who work for me in the office

- my finances

but I would cope with...

- anxiety attacks

- the noise on the street when I'm trying to study

- an irritating boss

- unexpected expenses

I hope this helps!