I think "Earth" and "the Earth" are used pretty much interchangeably.
In general, you do not use an article with a proper noun in English. As you say, we do not say "the Mars", nor do we say "the Fred Smith". But you DO use an article with nouns that are not proper nouns. We do say, "the planet", "the man", etc.
But then there are a few special cases. We seem to think of "Internet" as a proper noun. After all, there is and can be only one: if you made another huge computer network, it wouldn't really be "another internet", but "another big network that resembles the Internet". But we still put "the" in front of it: "I access the Internet from my home computer", NOT "I access Internet ...".
And "Earth". Sometimes we say "Earth", sometimes we say "the Earth". Perhaps it's because we're not quite sure if it's a proper name or a general noun, like there might be other earths but this is the one we live on, but then again this is our home planet, THE Earth.
Note we capitalize Moon also, but when speaking of our own moon, it's always "the Moon". We don't say, "Apollo 11 travelled to Moon", but "... to the Moon". That's a little less mysterious because we talk about other planets having moons. In a sense it's natural to say that our own of something is "the", like I've heard people refer to their house as "the house", as in, "I've got to go the house and pick up my golf clubs". But that doesn't quite explain it, as we don't capitalize the word in such cases. No one writes, "I went to the House to pick up my golf clubs".
As with many things in language, I think you just have to learn the special cases and exceptions. Sometimes there are historical reasons that may or may not make sense or apply any more, but that's how it is.