#CH_3 OH# and HCI both dissolve in water, but the dissolved HCI conducts electricity, while the dissolved #CH_3 OH# doesn't. Why?

The # HCl # is an ionic compound which is separated by the polar nature of water into H+ and Cl- ions. These charged particles conduct electricity.

ethyl alcohol is a covalent compound which is separated by the polar nature of water into separate molecules. These separate molecules are not charged and so do not conduct electricity.

Be careful about "black and white" statements like "this doesn't conduct electricity". In reality, a solution of methanol and water does conduct electricity, just to a MUCH lower extent than a solution of HCl in water.

HCl dissociates into #H_3O^+# and #Cl^-# ions in aqueous solutions, and it fully dissociates (which is why hydrochloric acid is a strong acid). Methanol in water also dissociates into ions,

#2CH3OH = CH_3OH_2^++ −CH_3O^-#

The self-ionisation constant of methanol will be very low, it will be only marginally different to that of water (which is about #10^(−14)#.

So the ions will be present and will conduct electricity in a methanol/water solution, it just does it to a very very small extent. But to say that they "do not conduct" electricity implies a conductivity of zero. It will not be zero, but it will be EXTREMELY small.