Questions about gravitational and inertial mass 
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*What differences between gravitational mass and inertial mass? I cannot tell the differences between them. 

*In history, which concept was put forward firstly? 

*Are there some experiments to prove they are identical?
 A: Re your first question: What's the difference between the five masses: inertial mass, gravitational mass, rest mass, invariant mass and relativistic mass? is indeed a duplicate, as Qmechanic says, and there are too many excellent answers to it to be worth attempting a fresh answer here.
Re the second question: mass only became a useful concept when Newton formulated his laws. I think he arrived at his laws of motion first, then his gravitational law, so the idea of inertial mass predates gravitational mass. But from this distance in history both concepts arrived at about the same time.
Re your third question: the classic experiment to demonstrate the equivalence of inertial and gravitationl mass is the Eötvös experiment. A number of even higher precision varients of this have been done over the years since, but Eötvös pretty much settled the question.
A: *

*Well, in brief, the Passive gravitational mass $m_P$, as named by H. Bondi in the 1957 (see C.Will, Theory and experiments in gravitational Physics), is the one that enters the definition of gravitational force, e.g. between a massive body M and a test particle of passive gravitational mass $m_P$:
$$F=G\frac{M\cdot m}{r^2}$$
or, more often, in the definition of weight force: $\mathbf{F}=m_P\mathbf{g}$, where $g\simeq 9.81$ is the well known gravitational acceleration on the Earth. The inertial mass $m_I$ is the one that enters the Newton's definition of force $\mathbf{F}=m_I\mathbf{a}$. Newton's equivalence principle, or Week Equivalence Principle (see Will), states that $m_P=m_I$, i.e. the free fall of different objects is equal, despite their internal composition and structure.

*I'm not sure about which concept was born first but I think that the concept of inertial mass, within the definition of force, as stated by Newton in his 1687 Principia, was developed first since his universal gravitational law is certainly more complex and profound and was born later.

*There are lots of Eotvos like experiments! Basically, they test the dependence of the gravitational attraction with respect to the composition and material different masses are made of. You can find the references in the WEP tests section in the wikipedia page of the Einstein Equivalence Principle, which includes the WEP.

