What's the difference between Laplace Force and Lorentz force? Currently we are studying Laplace force, when I searched, I found also the Lorentz force, and I'm lost right now, I tried to understand it but I could not !
$$F=I\vec l \wedge \vec B$$
I didn't understand the other one ! 
 A: Well the difference is simple;
Laplace force: is a force applied on a conductor traversed by an electric current, and it's in an uniform magnetic field. We apply the right hand rule to identify its direction. Its intensity is calculated as you said with the following law:
$$F=I\vec l \wedge \vec B$$
Lorentz force : is the force applied on an electric charge moving in an electromagnetic field, to indentify the direction we use again the right hand rule, and its intensity is given by :
$$F=q\vec v \wedge \vec B$$
The difference between these forces is obvious, the first one is applied on wire traversed by an electric current $I$, the second one is more general and we can observe that the electric current can be transformed to a movement:
Let's do some maths:
During $t$ the charge traversed in this wire is $q=It$ and the speed of these charges is  $\vec v=\frac{\vec l}{t}$
Multiplying $q$ and $\vec v$ we get :
$$q\vec v= I\vec l$$
Back to the Lorentz force; from the definition it's the force applied to the charges in an electric or magnetic field. When the conductor is traversed by an electric current the electrons are influenced by the Lorentz force, thus they can move the wire, $$F=q\vec v \wedge \vec B = I(\vec l \wedge \vec B)= I.l.B .\mathrm{sin}(\widehat{\vec l , \vec B})$$
to conclude :
The Laplace force is caused by Lorentz force, this one is applied to the charges in motion , and it causes the move of any wire in an uniform magnetic field. 
I hope that my explanation be helpful, good luck !  
