TL, DR : While you could say a MOBA is a RTS, it doesn't include a lot of things a typical RTS does. It's a real time strategy game but in the strict definition of the term and doesn't fit very well in the RTS games genre. Here I'm saying it's a game being play in real time needing strategy but it far from being a RTS type of game. Read below to understand my reasoning.
Multiplayer Online Battle Arena, aka MOBA, or action real-time strategy (ARTS), takes place in a pre-defined arena between 2, or more, teams.
The focus of a MOBA is to take down a central building (called nexus in Lol, Core in Heroes of the storm...) of the other team, which is within a base defended by various type of buildings (depending of the game). The usual arena include 3 "lanes" where minions go down in wave to the other team. They are variants with more or less lanes (Twisted Tree in Lol for example).
In a MOBA, you usually control one character for the whole game. That character gains experience by killing minions, neutral champions and enemy champions. With the experience, it levels up and gain more power and abilities. Their might not be neutral minions in all MOBA.
There's more to MOBA but that's the core of what makes a MOBA.
Real Time Strategy, aka as RTS, the focus of the game depends, usually, on the mission you have to complete. There's too much different missions possible to be listed here but, can include, gathering X amount of resources in a certain time, kill X amount of units, take control of Y locations on the map, clear the maps from all enemies, stay alive for Z minutes, etc.
In a typical RTS, you gather resources, build a base, build units and take on your mission. Some RTS focus on each of those 3 aspects differently even leaving one or two only to the player control. Some mission type also focus on one or two of those aspects only.
You can play a RTS alone, in single player mode which often includes a story mode and a player vs AI mode. You can also play multiplayer online or LAN.
You can play alone in a MOBA to practice but you'll never really play a MOBA game alone as a team includes more than one champion. In the past year (or 2), AI have been introduced in Lol (don't know about other MOBA) which makes it possible but it is mainly seen as practice than actually playing the game as defined by the MOBA genre. Some MOBA games offer LAN play, not all of them.
The focus of each game is obviously different. In a MOBA, the focus never change from a game to the other while in a RTS, the mission you play guides the focus.
There's many differences I've not covered but I think that's enough to see why both are different from each others.