The weights in an artificial neural network are an approximation of multiple processes combined that take place in biological neurons. Myelination plays a role, but not a major one. Weights in artificial neural networks can be positive or negative numbers.
Weight magnitude. The magnitude of a weight is analogous to a combination of increased dendritic connections between neurons, number of synapses between their dendrites, density of neurotransmitter receptors on the post-synaptic terminals, as well as increased neurotransmitter vesicle formation and fusion on the pre-synaptic terminals.
Positive weights. Positive weights are analogous to the pre-synaptic terminals of the synapses releasing excitatory neurotransmitters (i.e. glutamate). They make it more likely the receiving cell will fire an action potential.
Negative weights. Negative weights are analogous to inhibitory neurotransmitters (i.e. GABA) being released at the synapse. They make it less likely the receiving cell will fire an action potential.
Myelination increases the distance that action potentials can travel down the axon. The membrane voltage potential decays much closer to the cell body if the axon is not myelinated. One analogy is that of a garden hose. If the axon is not myelinated, then the garden hose is leaky with holes, and less water pressure (that caries water pressure waves, the action potentials) makes it to the end of the hose.
However, increased myelination is not necessarily analogous to an increased weight. For example, if there are few dendrites between two neurons, with few synaptic connections, then myelination would have little effect.
You also mentioned learning. During learning, the weights in an artificial neuron increase or decrease. In biological neurons, learning takes place at multiple scales and areas (i.e. non-synaptic and synaptic). At first, during increasing connection strength, you may see increased vesicle fusion, then increased neurotransmitter receptors, then new synaptic boutons, and new dendrites. When "unlearning", or forgetting, these processes happen in reverse.
Overall, a weight in artificial neurons clump a lot of biological complexity into one number that only crudely approximates the degree of connection strength between two biological neurons.