What is the difference between equatorial and ecliptical coordinate system? Of course, I know what the ecliptic is and what equator is. The problem is I still can not understand why we need two systems for measuring the same thing - are each of them useful only in specific occasions? For example, which is preferred in marine navigation? I am interested in this practical side of the question

Can you give an example which positions a star, using both coordinate systems?
To locate an object you need a system of coordinates. 

You need to define a plane (to define the latitude) and a direction (from which to measure longitude). On Earth, the obvious plane is the Equator. There is no obvious direction from which to measure longitude (which in the past led to different definition of longitude in France and Britain)

There is no single, natural plane in the sky. Several are possible: You can project the equator into the sky. You can use the plane of the Earth's orbit around the sun. You could use the plane of the galactic disc.

Using the Equator is simple, but rather Earth-centric. The Ecliptic system is more natural for solar system objects. The galactic system nice for deep space objects. The lack of a single natural plane in the sky means that several different coordinates systems can co-exist.

As an example Rigel, at RA 5h14'32'', Dec −08° 12′ 05''

Has a Lat-Long of:


-31.11°  76.09°   Ecliptic
-4.40°    219.66°  Galactic