What factors are responsible for making the molality and molarity values closer?

Answer 1

??? If you are refering to a specific experiment please give more details.
If you are asking in general, then let me rephrase your question.

When will molality and molarity value be almost the same?

Molarity = mole solute per L of solution
molality = mole solute per kg of solvent

Thus Molarity=molality when 1 L of solution corresponds to 1 kg of solvent. That would be true if:
1. There is no significant change in the volume of the solvent when you dissolve the solute and
2. If the density of the solvent is 1 g/mL (or 1 kg/L if you prefer).

So, for a lot of dilute aqueous solutions you have a negligible change in volume due to the solvation of the solute, thus Vsolution=Vsolvent. This means 1 L of solution is 1 L of solvent.
For water d=1 kg/L, so 1 L of water weighs 1 kg. But we said that 1L of solution is 1 L of water, thus 1L of solution corresponds to 1kg of water (solvent) and therefore M=m