At low/middle levels, weapon damage may be higher than the Martial Arts die
At lower levels, the monk's Martial Arts die is lower than most weapons' damage die. As the Monk Table shows, it starts at 1d4, goes up to 1d6 at level 5, 1d8 at level 11, and 1d10 at level 17.
By comparison, the monk weapon options for non-Kensei monks are shortswords and simple melee weapons that don't have the two-handed or heavy property. There are 4 simple melee weapons that do 1d4 damage, and 5 simple melee weapons as well as the shortsword that do 1d6 damage (the quarterstaff and spear also have the versatile property and can be used two-handed for 1d8 damage).
Way of the Kensei monks have a larger set of options; the Path of the Kensei 3rd-level feature (XGtE, p. 34) says:
Choose two types of weapons to be your kensei weapons: one melee weapon and one ranged weapon. Each of these weapons can be any simple or martial weapon that lacks the heavy and special properties. The longbow is also a valid choice. You gain proficiency with these weapons if you don’t already have it. Weapons of the chosen types are monk weapons for you.
They can pick a third kensei weapon at 6th level, a fourth at 11th level, and a fifth at 17th level. This adds many weapons with a higher damage die; besides the 1d4 and 1d6 damage weapons, it adds 9 weapons that do 1d8 damage (the battleaxe, longsword, and warhammer also have the versatile property and can be used two-handed for 1d10 damage).
The Martial Arts damage die doesn't catch up to the weapons that do 1d6 damage until 5th level, and it doesn't catch up to the weapons that do 1d8 damage until 11th level. It only matches the kensei weapon options that do 1d10 damage at 17th level. Thus, especially for Way of the Kensei monks, picking a high-damage weapon will leave you better off until it evens out at very high levels.
Different weapons have different weapon properties
Each weapon in the Weapons table has different weapon properties.
For instance, the dagger, handaxe, light hammer, sickle, and shortsword have the light property. The main interaction of the rules with the light property is the Two-Weapon Fighting rule; if a character is wielding a light one-handed melee weapon in each hand, they can attack with one as part of their Attack action and then use a bonus action to attack with the other (but don't add their ability modifier to the bonus-action attack). This is generally suboptimal for monks since they can just make an unarmed strike with their bonus action anyway, but it's a possibility.
Furthermore, the Martial Arts feature eliminates weapons with the two-handed or heavy property as monk weapons, and the Path of the Kensei feature allows you to choose any weapon that doesn't have the heavy or special property as kensei weapons (which are monk weapons) - but this doesn't rule out the versatile property. This means you can get a slightly higher damage die by choosing a versatile weapon - and you can still make unarmed strikes as your bonus action, since they don't require a free hand.
The impact of the thrown property is obvious - it lets you make ranged attacks without losing the benefits of the Martial Arts feature. This isn't quite as big a benefit for Kensei monks since they could just pick an actual ranged weapon as a kensei weapon if they wanted, but it still lets them have a single weapon that can be used in melee or at range.
(The finesse property is irrelevant to monks - they can use Dexterity instead of Strength for any monk weapon anyway.)
And finally, the reach property lets you have a reach of 10 feet with the weapon instead of 5 feet. There's only one such weapon that's relevant to monks, and only to Kensei monks: the whip. It's a martial melee weapon with the finesse and reach properties (all other weapons with the reach property are martial melee weapons with the heavy and two-handed properties).
Game features might interact with specific weapons differently
As Rykara's answer notes, the Polearm Master feat (PHB, p. 168) only works with certain weapons, two of which - the quarterstaff and spear (as of the 2018 PHB errata) - are monk weapons. The first benefit is useless to monks because it takes their bonus action to do 1d4 damage, which they can do as a class feature even at 1st level - but the second benefit, letting them make opportunity attacks when a creatures enters their reach, may be useful to a monk. (Still not a great choice for monks.)
Following on from the previous section, the Defensive Duelist feat (PHB, p. 165) says that, when you're wielding a weapon with the finesse property and you're hit by a melee attack, you can use your reaction to add your proficiency bonus to your AC for that attack. This only works with finesse weapons, so it matters what weapon you're wielding.
Magic weapons might have other effects
There are many kinds of magic weapons. Whether it's a shortsword +1, a dagger of venom, an oathbow, or a holy avenger, each magic item normally only appears in a certain form or forms. Each one has its own special effects, whether that's simply to do extra damage, cast light in a radius, or something else. Obviously, the effect varies depending on which magic weapon it is, so which weapon you wield matters in that regard.
