The odds are the same in both cases ie 1 in 36. If it helps imagine that when the two die are thrown together, one settles first and the second one wobbles about for a bit longer before settling - just like throwing one dice twice really !!
The first answer is correct if there is no bais in the dice. if there is a slight weight imbalance in the single dice this will increase/decrease the odds depending on the imbalance. This theory is used to try to predict lottery numbers, roulette wheels ect. although it has to be said you normally have to run these things for a substantial amount of time before any bias is discovered