Hall voltage/resistance is measured perpendicular to the current applied. Here current, magnetic field and the direction of measuring the hall voltage are mutually perpendicular to each other (strictly).

For measuring magnetoresistance,  electric current and measured voltage drop are parallel to each other and magnetic field can be applied perpendicular (transverse) or parallel (longitudinal) to the electric current.

The main difference between these two measurements is that the voltage drop is always measured perpendicular to the current for Hall voltage which is not for the case of magnetoresistance.

Hi, Rudra Prasad Jena, I am asked why Lorenz forces are suitable for Hall effect, and not suitable for Magnetoresistive effect.

Magnetoresistance IS related to a Lorentz force (https://en.wikipedia.org/wiki/Magnetoresistance) but the force itself isn't the resistance. Resistance is a loss of energy due to electrons (or more generally charge carriers) scattering off of defects in the conductor (at least in the case of metal) and losing energy in the form of thermal phonons.  When a magnetic field is applied to a conductor with a current, electrons are forced to move in orbits on the Fermi surface rather than propagating along the path imposed by the current.  Thus, for the same amount of "forward" progress of the charge carrier, it will have moved considerably further along an orbital path, thereby increasing the likelihood of scattering and losing the energy imposed by the applied field (voltage) forcing the current to flow.

By Michael Foegelle - "When a magnetic field is applied to a conductor with a current, electrons are forced to move in orbits on the Fermi surface rather than propagating along the path imposed by the current". Why this explanation is not suitable for Hall efect?

I'm not sure what you mean by "suitable".  Both effects can be explained through a discussion of the fact that a magnetic field will cause a charged particle to try to follow a curved path.  The Hall effect could be envisioned as a special case of magnetoresistance where the electrons are being scattered by the edge of the conductor carrying the current.  The induced potential is caused by charges in the current being redirected from their uniformly distributed equipoetential across the conductor.  The steady state is restored when the charge distribution on either side of the conductor counters the Lorentz force on the charges in the current.  However, from the standpoint of magnetoresistance, the edge of the conductor is not required.  In the example in the wiki article above using the washer arrangement, the Lorentz force is generating a current perpendicular to the current flow, but there is no "edge" for the charges to hit (they can go around the circle of the washer forever if nothing scatters them) but there's still a magnetoresistance term.   

Here is illustration of charges paths ih magnetic field. The Hall effect is explaind with the help of Lorentz forces. But Magnetoresistance effect conflicts with this explanation on angel dependence.

I'm still not quite sure what point you're trying to make, but those plots actually show an important point that I didn't really want to get into, which is charge orbit diameter (which is a function of the applied magnetic field).  If you envision the first picture as representing the case for the Hall effect, the open orbits intersect the side of the conductor before they can complete an orbit, thus the net potential due to the current flow.  However, if you envision the latter drawings as the case with higher field strengths, most of the charge will complete an orbit without ever intersecting the edge of the conductor.  Thus, they do not contribute to the Hall effect (except possibly at the surface), but they DO exhibit an increase in apparent resistance due to the longer path taken by the charge over that of the zero field case.  

Note too that there's the case of AMR, where the magnetic properties of the material come into play.  In that case, the localized field that the charges see is different from the field that is applied to the material.  

The wiki on magnetoresistance seems  quite complete in describing the theory, giving the result in terms of a reduction in carrier velocity, which is basically saying that the average velocity component in the direction of the current flow is reduced when the charges start moving in circles.  Remember, the actual velocity of a single charge in the electron gas of a conductor is essentially a constant, whether it's moving it its orbit around a single atom or under the influence of an applied field.  Charges just move to screen the applied electric field (cancel it out) and bend their path under the influence of a magnetic field.  

At any rate, I still don't quite see what conflict you're referring to.

By Michael Foegelle - "the average velocity component in the direction of the current flow is reduced when the charges start moving in circles". Thus the Charges are frozen by Magnetic field. And what happens with resistance in this case? De facto Resistance in Ferromagnetics decreases in this case. And this is strange.

No it's not if you take into account that ferromagnetic materials have domains of inherent magnetism that are restricting the current flow already.  The fact that an external field can align those domains in such a way as to either increase that resistance, or to cancel out the native effect and reduce that resistance, should not be surprising.

By Michael Foegelle - "external field can align those domains in such a way as to either increase that resistance, or to cancel out the native effect".  This is a good hypothesis. But can it explain the angular dependence of Magnetoresistance?

R=R0+dR*cos2(Ang)

There are 3 vectors: I-current; M-magnitization; H-external magnetic field.  Current I flow along X-axis. MX=0 -> Ang=90/-90; MX=M -> Ang=0; MX=-M -> Ang=180. And what is orientation of H relative to M ?

Case of canceling? Case of  incriasing?

Given I don't work in this area and these days for the most part don't even have to deal with solid state physics anymore, I'd recommend you take a look at some of the references on the subject rather than expecting me to give you an analysis of all the research done on the phenomenon to date.  However, if I look at the problem as a basic scenario where the non-applied-field state consists of a collection of randomly oriented domains, each of which has a magnetoresistance that is a function of their orientation relative to the current flow, and realizing that the magnetoresistance is itself a function of the angle between the field and the current flow, then it should be apparent that when an externally applied field causes the inherent magnetization to be along the current vs. when the angle is perpendicular to the current, that the two cases would bound the case where the domains were randomly oriented, correct?  

Regarding your equation above, if you go back to the Wiki I mentioned (https://en.wikipedia.org/wiki/Magnetoresistance) you'll see that the equation is slightly different, defined in terms of R_perp and R_||.  

R = R_perp + (R_|| - R_perp) * cos^2(angle)

What's missing from that is the relationship of B (and thus R) on the M and  H terms.  As mentioned in the Wiki, that's highly dependent on the type of material and thus is far beyond a simple analysis here.  Still, if I were to represent the above equation as 

R = R_perp + (R_|| - R_perp) * sum(cos^2(angle_i))/N

and compare the results of angle_i = 0, angle_i = pi/2, and angle_i = random, it should be apparent that the third case will be somewhere between R_perp and R_||.  Granted, not all materials will exhibit this behavior, and probably not at all levels of applied field, but this should make it clear as to how it COULD happen.