This is quite a basic question where you can easily find (plenty of) information in Google:

https://en.wikipedia.org/wiki/Molecular_dynamics

https://en.wikipedia.org/wiki/Monte_Carlo_method

Basically, molecular dynamics simulates molecular movements by solving the Newton's equations of motion for the molecules, whereas Monte Carlo calculates thermodynamical statistical probabilities of acceptance/rejection of moves.

Dear Kedharnath Arumugakani,

There are many things wrong about your answer:

Most fundamentally wrong is that you think that there are time scales in MC simulations: there is absolutely no time in MC simulations. A computation cycle costs almost the same for both methods (but algorithms are more efficient in MD).

Also what you say about the length is wrong. Distances scales are comparable in both.

FYI

The objective of a Monte Carlo(MC) simulation is to generate an ensemble of representative configurations under specific thermodynamics conditions for a complex macromolecular system. Applying random perturbations to the system generates these configurations. To properly sample the representative space, the perturbations must be sufficiently large, energetically feasible and highly probable. Monte Carlo simulations do not provide information about time evolution. Rather, they provide an ensemble of representative configurations, and, consequently, conformations from which probabilities and relevant thermodynamic observables, such as the free energy, may be calculated. Monte Carlo simulations are not only important on their own right, but they also play a fundamental role when designing complex and hybrid molecular dynamic (MD) algorithms.

Molecular dynamics studies the temporal evolution of the coordinates and the momenta (the state) of a given macromolecular structure. Such an evolution is called a trajectory. Atypical trajectory is obtained by solving Newton’s equations. The trajectory is important in assessing numerous time dependent observables such as the accessibility of a given molecular surface, the interaction in between a small molecule (e.g., a drug) and the hemagglutinin or the neuraminidase of a given influenza strain, the interaction epitope-paratope in between an antigen (e.g., hemagglutinin) and an antibody (e.g., CR8020), the appearance and disappearance of a particular channel or cavity, and the fusion of the hemagglutinin with a cell membrane (fusion peptide), amongst others. The choice of a proper potential is of the utmost importance in obtaining accurate molecular dynamics simulations. The potential must be physically sound as well as computationally tractable. An approximate potential may be calculated from quantum mechanics and from the Born-Oppenheimer approximation in which only the positions of the atomic nucleus bonding is considered. The potentials may be divided into bonding potentials and long-range potentials. The bonding potentials involve interaction with two atoms (bound lengths), three atoms (bound angles), and four atoms (dihedral angles). Long-range interactions are associated with the Lennard-Jones potential (van der Waal) and the Columbic potential. The harmonic approximation is utilised for the bonding potentials, which means that solely small displacements are accurately represented.

Cited from 2015 Molecular Dynamics, Monte Carlo Simulations, and Langevin Dynamics-A Computational Review

FYI:

Compare with molecular dynamics, Monte Carlo relies on statistical mechanics and it generates states according to appropriate Boltzmann probabilities, instead of trying to reproduce the dynamics of a systems. MC can be deal with problems with larger time and space scales than MD, and it has been used to simulate DNA flow through entropic trap array where polymer is modelled by a lattice model with bond fluctuation.

cited from 2015 Dissipative Particle Dynamics (DPD)-An Overview and Recent Developments

FYI:

Molecular simulations (molecular dynamics and Monte Carlo approaches) offer a description of matter at the atomic level. Therefore, they are useful tools to interpret many experiments covering spatial and time scales matched by neutron scattering in clay minerals. In molecular dynamics simulations (MD), the atoms are considered as classical objects submitted to the Newton's equation of motion, which is propagated step by step thanks to an algorithm (usually Verlet or Velocity Verlet algorithm). In Monte Carlo simulations (MC), the accessible configurations of the phase space are sampled according to the Metropolis algorithm. In both cases, the interactions between the atoms must be calculated using a force-field, which gathers all the atomic parameters necessary for this calculation. Most often, the interaction between two atoms is taken as the sum of an electrostatic interaction, a van der Waals attraction and an interatomic repulsion, which depend on the distance between the atoms. From the trajectories obtained with MD, structural and dynamical quantities like densities, radial distribution functions or diffusion coefficients can be calculated from mean squared displacements, but also scattering functions, which depend on the relative positions of the atoms. In MC, dynamical information is not accessible. However, contrary to MD, MC allows simulations using statistical ensembles such as the grand-canonical ensemble, where the number of particles in the system can fluctuate.

Cited from Neutron scattering, a powerful tool to study clay minerals

FYI:

There are two types of molecular simulation, Molecular Dynamics (MD) and Monte Carlo (MC) methods.

The Molecular Dynamics method shows the time evolution of particles by numerical integration of Newton-Euler equations of motion. At each step, positions (coordinates and orientations) and velocities (linear and angular) of the particles are calculated. The forces are derived from the potentials acting between particles (i.e., force field). MD can be used to study both equilibrium and time dependent properties of the system.

In the Monte Carlo (MC) simulation method particles are moved randomly, to represent a target probability distribution consistent with the desired state of the system. As a result, MC is not deterministic (i.e., the system does not evolve following a physical path and real trajectories cannot be generated using MC method). Therefore, MC can be used to study properties of the system in thermodynamic equilibrium.

The Molecular Dynamics (MD) method is a deterministic technique that describes the time evolution of a system of particles by solving Newton (for spherical particles) or Newton-Euler (for non-spherical particles) equations of motion. The first step is to create initial condition that includes coordinates and orientations of particles. Also the force field which is the key driver of the MD simulation is defined. A single particle is affected by the potential energy of the other particles in the system. The force field defines a set of functional forms that are used to calculate the potential energy of the system based on the relative position of particles. More explanations and examples will be provided later. After defining run parameters, the system energy will be minimized. Initial energy minimization finds a configuration corresponding to a local energy minimum. If in some parts of the model particles are too close to each other, the forces may become very large and the MD simulation may crash. This step eliminates hot spots of the initial configuration. The next step of the simulation is carried out by solving Newton-Euler equations of motion. All dynamic simulations are performed in two stages: equilibrium and production. In the equilibration stage, the initial random structure is evolved until a steady state is reached. The purpose of this stage is to prepare the system to sample the phase space from the desired probability distribution consistent with the ensemble of the simulation. As the system becomes larger, the amount of time that is needed to equilibrate the system increases since it should explore larger phase space. The state of the system is evaluated by calculating and plotting different thermodynamic variables such as energy, pressure and temperature. When the system reaches the steady state, the state variables will fluctuate around a stable average value. After equilibrating the system at the target ensemble data can be collected for the production stage. The length of the production stage depends on the type of the property being calculated and the size of the model. Plotting running average of the properties is a way to check the convergence of the result. The methods used for the equilibration and production are the same. Just we need to give the system some time to equilibrate and then we can start sampling and calculating properties. For instance, to keep the temperature at T=300 K, we have to modify the Newton-Euler equations of motion. The physical interpretation is to put the system in contact with a heat reservoir. It takes some time for the system to reach thermal equilibrium with the reservoir. Numerically this is done by the exchange of momentum between particles in the system and the reservoir through solving modified Newton-Euler equations of motion. With the language of statistical mechanics, this means we give some time to the system in order to evolve in the phase space (i.e. to visit different states) until the system starts to sample from a specified probability distribution associated with the ensemble of the simulation (i.e. generate certain states consistent with the ensemble).

In the MD approach statistical ensembles with probability distributions for target ensemble were obtained by solving Newton-Euler equations of motion. Monte Carlo (MC) is an alternative simulation method. In the MD method, the positions and velocities of particles are updated in each time step to generate ensemble of configurations. In contrast, sampling for the MC method depends on positions only (not velocities). In fact, the contribution of kinetic energy to the partition function is integrated analytically as its functional form is known.

cited from Multiscale modeling of clay-water systems

Dear Ma,

You know you can edit your answers and just put the link instead of flooding the space with irrelevant text?

Dear Fernando

Yes, you are right, I should conclude the ideas of other researchers, instead of just copying and citing the title of paper.

Cheers

There are many textbooks about the subject (literature about is vast), but in fact: What do you want to calculate? Systems in equilibrium or not?

FYI:

Article The role of molecular modeling in confined systems: Impact a...