In short, γῆ relates to part of the physical earth and sometimes the whole planet - with or without man, whereas κόσμος usually is used in the context of mankind. As you note, γῆ represents the Hebrew אֶרֶץ (e.g. Genesis 1:1 LXX). It is confusing in English, because the English congnate "cosmos" doesn't really mean the same thing as the underlying Greek word. Roughly, γῆ means "earth" or "land", whereas κόσμος means the "world" [of men].

Examples of γῆ in the New Testament, for instance:

Matthew 2:6

And thou Beth-lehem, in the land of Juda, art not the least among the princes of Juda: for out of thee shall come a Governor, that shall rule my people Israel.

^{Καὶ σὺ, Βηθλεέμ γῆ Ἰούδα, οὐδαμῶς ἐλαχίστη εἶ ἐν τοῖς ἡγεμόσιν Ἰούδα· ἐκ σοῦ γὰρ ἐξελεύσεται ἡγούμενος, ὅστις ποιμανεῖ τὸν λαόν μου τὸν Ἰσραήλ.}

Matthew 5:18

For verily I say unto you, Till heaven and earth pass, one jot or one tittle shall in no wise pass from the law, till all be fulfilled.

^{ἀμὴν γὰρ λέγω ὑμῖν, ἕως ἂν παρέλθῃ ὁ οὐρανὸς καὶ ἡ γῆ, ἰῶτα ἓν ἢ μία κεραία οὐ μὴ παρέλθῃ ἀπὸ τοῦ νόμου, ἕως ἂν πάντα γένηται.}

Examples of κόσμος:

Matthew 4:8

Again, the devil taketh him up into an exceeding high mountain, and sheweth him all the kingdoms of the world, and the glory of them

^{πάλιν παραλαμβάνει αὐτὸν ὁ διάβολος εἰς ὄρος ὑψηλὸν λίαν, καὶ δείκνυσιν αὐτῷ πάσας τὰς βασιλείας τοῦ κόσμου καὶ τὴν δόξαν αὐτῶν}

John 3:16

For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life.

^{Οὕτω γὰρ ἠγάπησεν ὁ Θεὸς τὸν κόσμον, ὥστε τὸν υἱὸν αὐτοῦ τὸν μονογενῆ ἔδωκεν, ἵνα πᾶς ὁ πιστεύων εἰς αὐτὸν μὴ ἀπόληται, ἀλλʼ ἔχῃ ζωὴν αἰώνιον.}

κόσμος is used 258 times in the Greek Bible (New Testament plus Septuagint) and γῆ over 3,300 times, so there are bound to be counter examples. But I think in general the usage falls along the above lines. I think, though, that although one might see γῆ being used in place of κόσμος to describe the "world of men", there are few if any examples in the other direction.

(I think this answer can be improved upon. I didn't really pay any attention to the Hebrew words that κόσμος translates in the Septuagint.)