Difference between simultaneous and ordered N.E. in the problem. Consider the following game. There are two players and one of them could be in two states:
$A$ with probability $p$ and $B$ with probability $1 - p$. Both players actions are $1, 2,$ and to not participate.
If the state is $A$ then players get $\$1$ if 1 is majority action, 1/2 if they tie, and 0 if 2 is the majority. Likewise, we determine the game in the state $B$.
Now we have a question to find Bayes-Nash equilibrium (pure) in two situations: (1) simultaneous game and (2) first go first, and then second take part.
I've decided that in first case we have: N.E. not participate for first player and $(1 , 2)$ (say $1$ in situation $A$ and say $2$ in situation $B$). However I've got the same for simultaneous game. There is a problem?
 A: Your description of the game is a bit vague, so I have to make a few assumptions. First, you say "Likewise, we determine the game in the state $B$." I am assuming here the payoffs are flipped (payoff of 1 iff both choose "2"), otherwise payoffs would be state independent and there is no point in using Bayes Nash equilibrium.
Second, in the sequential game, I assume the actions of the first moving player is observable for the second (otherwise it would be equivalent to a simultaneous move game).
Third, you say "one of them could be in two states". It can make a difference which player observes those states and which player moves first in the sequential game. Suppose player 2 is the one who privately observes the state, but player 1 doesn't, and player 2 moves second. Then this situation is identical to the simultaneous move game. However, if player 1 were to move second, then it changes things, because the action of player 2 could then "reveal" the state to the other player.
Regarding equilibrium: You are right that the informed player choosing $(1,2)$ and the uninformed choosing "no participation" is a BNE. Any deviation by the informed player would strictly reduce his payoff, given the uninformed player abstains. And a deviation by the uninformed player cannot improve things but can decrease his payoff (if the incorrect action is chosen which does not match the state). You are also right that in the sequential game this is a BNE as well - after all, both players get the maximum payoff in every state with probability 1, so there is no way to improve on this.
There are additional equilibria in some cases. For example, in the sequential game, if player 1 moves first and is informed, then the following strategy is also a BNE: $(1,2)$ for player 1 and for player 2: 1 if first player chose 1 and 2 if first player chose 2. The outcomes in this equilibrium are the same, however, the uninformed player does not actually abstain. This equilibrium does not exist if the order of moves is reversed; this is why I pointed out above it might matter.
