You are probably drinking vodka with a higher alcohol content than the rum. Look on the bottle to see the percentage of alcohol content of each product you are drinking. Also, are you using the same amounts of each when mixing your drinks? How much you put in each drink would also be a major factor.
How drunk you get when drinking one drink versus another is dependent upon the proof of each drink. If the rum is say 50% proof and the vodka is 100 proof it means that there is considerably more alcohol content in the vodka which is why it makes you more drunk than the rum does.