Whether it's important or not is entirely irrelevant. It's capitalized if it's a proper noun, name, or nickname, it's not if it's just a regular noun or word. Consider this sentence: "I love you, Dad, even though you're not my real dad." In the first usage, it's being used as a nickname or title. The second time, it's just a plain noun. It's specifically the context in the sentence that dictates whether it should be capitalized or not.
The Bible does the same with the word "god". In many cases, it's used as a proper noun, as in "And God saw that it was good." but when it's not, it's not capitalized, as in "Thou shalt have no other gods before me". Heck, if you want a more clearly related example, look at the word "earth". Typically, when it's preceded by "the" it's not capitalized, such as "After years of nuclear war and destruction, the earth was barren and the land no longer fertile", but when used as a name it is, "He got tired of living on Mars, so he went back to Earth."
For "the planet" you have a bit more freedom. If it were used side-by-side with a named planet, as in "Mars, Venus, and The Planet", it would definitely be capitalized, but in every other context it's just a matter of whether you consider "The Planet" to be an object or a name. Lowercase is probably better, but it certainly wasn't wrong in the original translation either (unlike JENOVA, which was obviously wrong).
tl;dr - This specific case is a matter of preference. Neither is wrong, but all lowercase is probably a little more right.