Well, a "why" question can be tricky if you're asking about why the designers chose to do things that way, as unless you can get a quote from them on the process and reasoning they used we're just speculating.
What's does the book say?
But, presumably the designers thought it was somewhat clear enough to make them separate skills. So, let's take a look at the differences between Deception and Persuasion, and see if we can draw as clear a line as we can between them. Quoting from the online Basic Rules, "Using Ability Scores", "Using Each Ability", under "Charisma Checks":
Deception. Your Charisma (Deception) check determines whether you can convincingly hide the truth, either verbally or through your actions. This deception can encompass everything from misleading others through ambiguity to telling outright lies. Typical situations include trying to fast-talk a guard, con a merchant, earn money through gambling, pass yourself off in a disguise, dull someone’s suspicions with false assurances, or maintain a straight face while telling a blatant lie.
Persuasion. When you attempt to influence someone or a group of people with tact, social graces, or good nature, the DM might ask you to make a Charisma (Persuasion) check. Typically, you use persuasion when acting in good faith, to foster friendships, make cordial requests, or exhibit proper etiquette. Examples of persuading others include convincing a chamberlain to let your party see the king, negotiating peace between warring tribes, or inspiring a crowd of townsfolk.
For Deception, phrases that stick out to me are "hide the truth", "mislead others through ambiguity", "pass yourself off in a disguise", and "maintain a straight face".
For Persuasion, phrases that stick out to me are "tact", "good nature", "foster friendships", "proper etiquette", and "inspiring a crowd".
So how do I tell which one to use?
Sure, I can see that there could be some overlap in some cases and can certainly understand your concern. But I think the key is to focus on what you're doing and the skills you'd need to do them, not solely just on what you're saying or whether what you're trying to convince somebody of is the truth.
Are you trying to invent a plausible reason for the guard to let you in? Sounds like Deception.
Are you trying to put the guard at ease for the guard to be able to trust you? Sounds like Persuasion.
Are you trying to disguise yourself as another guard or as a noble? Sounds like Deception.
Are you trying to have the guard have a stronger sense of duty to the king? Sounds like Persuasion.
Any of those actions could be taken in an encounter with a guard, regardless of whether the reason you're trying to talk to the guard is entirely truthful. And an encounter with the guard could quite reasonably involve multiple actions and therefore multiple checks, as you need to find which guard seems most friendly or distracted (Wisdom/Insight), make friends with him (Persuasion), and come up with a plausible reason for the guard to let you in (Deception). Each interaction that could have interesting consequences for the story (based on whether it succeeds or fails) could have its own check, and those checks should help you tell your overall story.
I'm getting a little sidetracked, but for more information on how social encounters can have multiple checks and be more interesting than saying "I throw a Deception at the guard" instead of "I throw a Dagger at the guard", see How to challenge a pacifist party as well as The Angry GM's Systemic InterACTION.
What if either Deception or Persuasion could work?
Even after all that, it's quite possible that a player could take an action where the DM is really unsure which skill proficiency should apply, or it seems that either could apply equally well. In that case, the DM could reasonably just allow the player to apply their proficiency if they have it in either. Remember, there are no skill checks, just ability checks. From the general Ability Check rules (with my emphasis added),
For every ability check, the DM decides which of the six abilities is relevant to the task at hand and the difficulty of the task, represented by a Difficulty Class.
Sometimes, the DM might ask for an ability check using a specific skill—for example, “Make a Wisdom (Perception) check.” At other times, a player might ask the DM if proficiency in a particular skill applies to a check. In either case, proficiency in a skill means an individual can add his or her proficiency bonus to ability checks that involve that skill. Without proficiency in the skill, the individual makes a normal ability check.
So, when a player takes an action interacting with the guard, the DM can just call for a Charisma check, and the player can (and should) say "Does my Persuasion proficiency apply?", and the DM can make a reasonable ruling on that question by itself, even if he'd answer the same way if asked "Does my Deception proficiency apply?". And certainly the DM could shortcut this, and just say "Roll a Charisma check, and you can apply either Deception or Persuasion". This can be part of a general "When in doubt, rule in favor of the players" approach that many DMs already have.