Q: Why does player X have a higher DVOA, but player Y has higher DYAR? What does each really mean, and which is more important when evaluating a player?
A: The easiest way to remember this is, DVOA is a rate stat, while DYAR is a cumulative stat. That is, DVOA aims to show how a player performs on a per play basis, while DYAR adds up the total contributions for a whole season (or game, drive, etc.). To compare to standard NFL stats: DVOA is similar to stats that measure per attempt (yards per carry, completion percentage), while DYAR is similar to total stats (yards, touchdowns, points).
A high DVOA signifies that when a player is involved in the play, the outcome is typically good, above league-average expectations. A high DYAR signifies that a player contributes to his team's success regularly, either through very good plays or a lot of mediocre ones (for example, giving 250 carries per year to a league-average running back is better than giving those same carries to someone off the practice squad). When players have a high DVOA but relatively low DYAR, it generally means they aren't "involved" in as many plays as their peers (e.g., No. 3 receivers, goal-line tight ends, Carson Palmer). When players have high DYAR but low DVOA, it generally means they are involved in a lot of plays, but haven't produced quite as much on each specific play.
As for which is more important, they really can't be played against each other like that. Both give insight into a player's contributions, and along with other stats (like RB Success Rate) can give a good picture of a player's performance. Remember also that these stats are all dependent on teammates (How is the line blocking? How good are the quarterback's receivers?), so we can't simply compare DVOA/DYAR to tell who is "better." These stats are a valuable tool to aid in comparisons, but do not completely replace observation.