天然 : 天 means Heaven, a vague reference to the Owner of the Universe; 然 means to cause to happen or brings about the existence of something or someone; so 天然 could mean created or the opposite of 人為 or 人造 。
自然 : 自 means self; 然 means to cause to happen or brings about the existence of something or someone; so 自然 means to happen naturally with additional participation of intentional beings or the Maker; about existence, it literally means self brings about its or his or her own existence which is logically impossible, it does not apply to existence, not even a God who is ever-existing 常在 or ever-living 常活 without a beginning or an ending. (永在 is not specific enough because it could mean with a beginning but without an ending.) I pray that the sharing of my grasp of these Chinese terms make sense to you :)