Depending on its context, אֶרֶץ can be translated as ground, earth, land, piece of ground, territory, country, region, earth, or underworld.1 It's a very common word. This is not to say it can be translated as any of these in any context, the context (esp. specific phrases in which it's used) guides how it should be understood.
Below is a visual representation of the various 'senses' carried by אֶרֶץ:2
The sense chart shows the various senses along with their frequency of use in the Hebrew Bible. It is clear that the senses of 'land' (whether in the 'sense' of being a nation or territory) and 'earth' are the most frequent senses implied by this word throughout the Hebrew Bible.
אֲדָמָה is a near-synonym (plesionym) of אֶרֶץ. It can be translated as earth, arable ground, soil, territory, world, or underworld, depending on its context. Another sense visualization makes this clearer:2
The evidence for synonymy comes from early translations as well. The Septuagint translated both terms as γῆς in this passage, which has a similar meaning in Greek.3
Given that these are synonyms yet different words, many translators use two separate words to prevent redundancy and to show this distinction. Even so, it could easily be translated,
... and a stream/mist would go up from the earth/ground and water the whole face/surface of the earth/ground....
Given that separate words were used by the original author of the text, it is likely that the slight difference in nuance expressed by most English translators is justified, for instance:
... but a stream would rise from the earth, and water the whole face of the ground... (NRSV).
But the best explanation is wordplay. In Genesis 2:7, the word for man/mankind is אָדָם (adam), and the connection between אָדָם and אֲדָמָה (adamah) is likely plain even for someone who doesn't read Hebrew. The sense is that 'man' (אָדָם) is formed by God from the soil/dust (אֲדָמָה), human life thus being described as a combination of a body formed from soil from the ground and breath given by Yahweh God (יְהוָ֨ה אֱלֹהִ֜ים), becoming a נֶפֶשׁ (nephesh), a living being.
So we see that in this creation account (beginning in 2:4), Yahweh God (יְהוָ֨ה אֱלֹהִ֜ים) made the earth (אֶרֶץ) and the heavens. However, a stream of some sort would rise from the earth (אֶרֶץ) and water the entire surface of the ground (אֲדָמָה, adamah). And then Yahweh God (יְהוָ֨ה אֱלֹהִ֜ים) formed 'man' (אָדָם, adam) from the soil/dust of the ground (אֲדָמָה, adamah).4 This wordplay seems most likely.
1 Ludwig Koehler et al., The Hebrew and Aramaic Lexicon of the Old Testament (Leiden; New York: E.J. Brill, 1999), 90–91.
2 Generated using Logos Bible Software v6.0a.
3 "πηγὴ δὲ ἀνέβαινεν ἐκ τῆς γῆς καὶ ἐπότιζεν πᾶν τὸ πρόσωπον τῆς γῆς." Septuaginta: With Morphology, electronic ed. (Stuttgart: Deutsche Bibelgesellschaft, 1979), Ge 2:6.
4 A creation account found in Nippur begins by observing that such a stream was not yet flowing. The Sumerian myth of Enki and Ninhursag also mentions such a watering system. Also, this idea of man being formed from the soil/dust of the earth is found elsewhere in Near Eastern mythology (e.g. the Atrahasis Epic explains the creation of man as a mixture of blood from a slain deity with clay).