The kinetic product is the product associated with the lowest energy barrier to its formation (the rate of product formation). The thermodynamic product is the most stable product with the lowest energy (the relative stability of the products)
The distinction is relevant when product A forms faster than product B because the activation energy for product A is lower than that for product B, yet product B is more stable. In such a case A is the kinetic product and is favoured under kinetic control and B is the thermodynamic product and is favoured under thermodynamic control as seen in the attached scheme.
At low temperature, the reaction is under kinetic control (rate, irreversible conditions) and the major product is that from fastest reaction.
At high temperature, the reaction is under thermodynamic control (equilibrium, reversible conditions) and the major product is the more stable system
Thermodynamic stability, on the other hand, is strictly a function of the change in free energy (ΔG), which is a state function, meaning that its value is determined exclusively by the difference between the initial state and the final state (or the free energy of the initial products and the final products). In other words, thermodynamic stability is totally independent of the pathway between reactants and products.
The classic textbook example is the conversion of diamond to graphite, which is thermodynamically favorable because the free energy of graphite is lower, but doesn't occur under ordinary conditions because the kinetics of the reaction (in the form of the immense activation energy required) are extremely unfavorable.
In summary, thermodynamic stability (in terms of free energy difference between reactants and products) will determine whether a given reaction could be theoretically spontaneous, but kinetic factors will decide whether the reaction occurs in practice under the given environmental conditions, as well as which pathway for a given reaction takes primacy.