Without wanting to go to much into the difference between task and conflict resolution, I'd say it's mainly a matter of scope. A conflict can be resolved with one or more tasks. For example, "Reducing my brother's pride" is a conflict that can be resolved using a ridicule task, "defeating the evil overlord" is a conflict that is likely to require several tasks, or possibly even sub-conflicts.
Most games have task resolution and let conflicts take care of themselves. Each PC has a skill package and the system resolves attempts to use these skills. Conflicts are resolved when one side is defeated. Because in most games the only measure of status is Health or Hit Points, defeat can only utimately be brought about by killing or physical restraint.
The usual defence here is that players don't like to be told what their character is doing so being persuaded by an NPC through mere skill use rather than failing to detect a lie, or a big bribe, is seen as anathema to the concept of player character. Some games, such as Dying Earth, do have explicit mechanisms for the task of persuading PCs but there isn't always an mechanism for enforcing the results. This can lead to some sulking amongst players.
Other games have an explicit conflict resolution mechanism. For Dogs in the Vineyard, the rule is that GM should agree with PC actions, and when he doesn't a conflict starts. This conflict is resolved thtough escalation from talking to fisticuffs to guns, if necessary. Conflicts are almost always of a moral nature and the stakes are decided up front. It could be "I persuade him to quit drinkin'" against "He whoops my sorry ass", or "I follow his trail" against "I become hopelessly lost and lose face with the Mountain Men". Dogs, as such, does not have a task resolution mechanism. Capes is a game in which all player interaction is through creation and resolution of conflicts. But again, there is no task resolution.
Various games have ways of changing the scope of a contest although these systmes are not explicitly for moving between task and conflict. Hero Wars differentiates between simple and extended contests by introducing APs (mechanically these change the distribution shape of likely rather than changing the likely outcome). The Shadow of Yesterday has "bringing down the pain" which works in a similar way. Both systems focus on the resolution of a contest with more detail than in usual to highlight the importance of the contest in the game. TSoY also has an added rule that this is the only way that you can get rid of a major NPC. It's probably the same in most instances of HW, it's just not explicitly stated.
To make games exciting it could be said that there needs to be a decent chance of success without eliminating the possibility of failure. A purely narrative point of view might also suggest that failure is just as interesting as success. In any case, what's important in the game is player involvement. So in task resolution games you don't want to eliminate characters early on (and usually there's plenty of healing around to deal with this problem) and in conflict resolution games you don't want the scope of the conflict to be so big as to resolve the whole game in one short conflict.
At this point it's probably sensible for me to note that I've opposed narrative games with task resolution which, to my mind, means that I don't think Hero Wars is a narrative game. Which I don't, so that's OK. My evidence is that it not only employs task resolution but that also there is no requirement for any kind of explicit stake or premise in a contest, except for nebulous states of victory or defeat (and in my experience of HW, they do tend to the rather nebulous).