Can someone PLEASE explain to me the interaction between the Monk's Martial Arts ability and Thrown Weapons.

According to Martial Arts:

You can use Dexterity instead of Strength for the attack and damage rolls of your unarmed strikes and monk weapons.

According to the Thrown modifier on weapons:

If a weapon has the thrown property, you can throw the weapon to make a ranged attack. If the weapon is a melee weapon, you use the same ability modifier for that attack roll and damage roll that you would use for a melee attack with the weapon. For example, if you throw a handaxe, you use your Strength, but if you throw a dagger, you can use either your Strength or your Dexterity, since the dagger has the finesse property.

Yet, everywhere I see on the internet, people say that Monk's can only use daggers for throwing 'cos they have finesse. A Handaxe/Javelin/Light Hammer/Spear are all monk weapons (simple melee weapons), they're just being thrown.

So these are my questions:
Can't you use Martial Arts to replace the STR mod with DEX on a thrown weapon when you throw it?
Why does finesse play a role when it's the same as Martial Arts?

To answer your questions, yes and it doesn't unless you're a rogue.

Can't you use Martial Arts to replace the STR mod with DEX on a thrown weapon when you throw it? Yes, as long as the weapon you're using is a Simple Melee Weapon (i.e. A monk weapon). So yes, they're able to use a Handaxe, Jaevlin or Spear. However, you want to use the base weapon damage until lvl 5, when your martial arts die increases to 1d6. But you can add your Dex modifier to any of those.

The reason that the item must be a Simple Melee weapon is due to the common mistake people make with the Darts that Monks come with. Their damage never changes from the base 1d4, but you do get to add your Dex modifier to it because it is a finesse weapon. This is because it is a Simple Ranged Weapon.

Why does finesse play a role when it's the same as Martial Arts? Finesse plays a role for those classes that are not a monk. For instance, a rogue would want a finesse weapon over a non type so they can add their Dex modifier rather than Str. Essentially, they needed a way for someone to be able to use either str or dex depending on what they wanted for some weapons.

Hopefully that helps!

Quote from Belx >>

The reason that the item must be a Simple Melee weapon is due to the common mistake people make with the Darts that Monks come with. Their damage never changes from the base 1d4, but you do get to add your Dex modifier to it because it is a finesse weapon. This is because it is a Simple Ranged Weapon.

 Darts are an odd case, you actually can use your dex modifier because it's a ranged weapon and you can use strength because it's finesse.  Doesn't really make a difference for them, but it's something to keep in mind if you decide you want to make a magical finesse longbow that lets you use strength.

So these are my questions:
Can't you use Martial Arts to replace the STR mod with DEX on a thrown weapon when you throw it?
Why does finesse play a role when it's the same as Martial Arts?

yes you can throw hand axes etc with Dex as it is a monk weapon. 

it doesn’t unless you are talking about non Monk classes.  

 So here's what the consensus is:

1. As a Monk, I can use DEX for a thrown weapon, so long as it's a Simple Melee Weapon (i.e. Monk Weapon). 
2. Finesse on thrown weapons is only important for non-Monks wanting to use DEX over STR. 

Thanks for your help with this guys. Looks like my dream for a Dwarven Monk throwing around Hand-axes is becoming a reality.

Also: Darts don't count as Monk Weapons 'cos they're Simple Ranged Weapons, not Simple Melee Weapons.

Monk Player "Hi DM I get these darts as a level 1 Monk, but I'm not allowed to use them as a Monk weapon. Is that right?"

DM "Yes"

Monk Player "But I could use a Dagger as a monk weapon which has, almost, exactly the same stats?"

DM "Yes"

Monk Player "And I could reskin those daggers to shurikens just like I could reskin a mace to some nunchucks?"

DM "Well I would say that shuriken stars would be more of a martial weapon, so no"

Monk Player "Could I at least have more daggers for my starting equipment instead of the darts?"

DM "No, if it's not in the RAW (rules as written) i'm not going to do it"

Monk Player "but I'm playing a Monk, haven't I been punished enough?"

DM "Oh dear, I hadn't thought of it that way. Here, have six magic +2 daggers reskinned as shuriken stars.........and a bag of holding so the party will like you."