Sometimes (and this is particularly true of low level adventuring parties) the dangers one faces are just too scary and too powerful to deal with on your own. Perhaps you simply cannot come up with a plan to assault a campfull of orcish raiders in the hills, or the scheming machinations of your political foes have procured a number of assassins whom you are rightfully afraid will murder you in the night. What do you do in these cases? Well, one option is to turn to the local authorities. Of course, since this is a fantastic medieval setting, they're not going to respond like the police would. They'll respond as feudal overlords and take immediate action to protect rights, land, and property that they feel are imperiled. If they're generally good, maybe they'll feel bad for anyone who might be hurt by the danger your party is incapable of facing, as well.
Either way: let's say you drum up a corps of knights to go kill those orcs. That is a solution to the problem, no doubt. But as a game where consequences matter, it's a solution that most players will find immensely unsatisfying for several reasons. The first is that they are adventurers. If they wanted to just beseech the local lord every time some ravening orcs were around, they wouldn't have gone out to adventure. They'd have remained farmers, masons, wheelwrights, whatever their parents were or their masters or what-have-you. Clearly, since they're the type to scrimp and save and then go out and buy a sword (a big deal in a medieval setting!) they have the impetus to stick those problems in the gut with two feet of steel.
The second reason is the natural response of authorities to adventurers everywhere: pure mistrust shading all the way to outright hatred. After all, adventurers present a locus of power that essentially challenges the authority of the local lords. Except in the wildest of frontiers, theirs is a power that rests on the ability to do exactly what adventurers make it their stock and trade to do: protect, defend, and slaughter. If someone sees an adventuring party as a legitimate source of justice or protection, it certainly undermines the position of their baron or count. Of course, the situation may exist where the baron or count is the one who hired the adventurers in the first place to act as his agents, which means he either lacks the means to address the problem in another way or (more likely) he will be incensed that his original investment of time, energy, and perhaps money has not panned out.
The third reason has to do with agency. PCs who are trying to solve a problem (such as that orc camp) have complete freedom to cook up any plan they want. Once some belted knights and a few hastily drawn up levees arrive, that game has essentially ended. They've surrendered their free agency in the face of a threat they cannot handle and have instead handed over the reigns to an authority figure. Now, they may be fine with this. But it means they have intentionally sidelined themselves and are now going to be taking orders, if they participate at all.
The last reason is the one that the characters will actually feel the most—the wallet. Their purse will be quite a bit lighter if they were hired to perform a job which they discover they are unable to perform. That intangible reward of local acclaim will also be far lower, one may wager.
Now, that's not to say that players should feel they can never ask anyone for help. After all, surviving and living to go on another adventure is a type of victory too. However, in the games I run actions have consequences. The natural consequences of passing off responsibility for a threat to a higher authority is that the rewards reaped are generally lower and the control of the operation usually passes out of PC hands.