What's the difference between these sentences?

Solution 1:

Suppose we pick some slightly less awkward examples:

He walked out, Rachel's words indelibly etched into his memory.

vs.

He walked out, Rachel's words being indelibly etched into his memory.

Still awkward, but I think both are grammatical.

Grammatically, I can't tell you anything certain, as my copy of CGEL is in my other pants. So I will speculate. In both sentences, everything after the comma is a dependent clause that's a variation of “Rachel's words were indelibly etched into his memory”. The first sentence is similar to “He walked out, his face red” or “his grey eyes flashing” or “the spell broken at last” or “the severed limbs of his enemies scattered behind him”. This sort of dependent clause is definitely grammatical, but I don't know what it's called. The second sentence is just a slightly different variation of the same clause, but again, I don't know what kind.

Semantically, there is just a slight difference. The first sentence only says that he walked out with Rachel's words etched into his memory. The second suggests that he walked out because her words were etched into his memory.

Update – Yep, they're clauses. Here, according to CGEL, they're being used as supplements. That means they're not grammatically required or tightly grammatically integrated into the sentence like a subject or object. They're tacked on at the end.

Example 1 contains a verbless clause. A plain old subject + predicate clause with no verb can be the object of the preposition with (“His pants (were) around his ankles” → “He was caught with his pants around his ankles.”) or, sometimes, a supplement (“He stormed out, his pants around his ankles.”)

Example 2 contains an -ing clause (what CGEL calls a gerund-participial clause). These are most often used in places where noun phrases could be used (“His death was not part of the plan” → “John being dead was not part of the plan.”), and the subject of the -ing clause is usually omitted (“Being dead was not part of the plan.”), so your example is unusual. Still grammatical, though.

Solution 2:

They are still awkward after your edit, and Robusto's adjustment would be to

He walked out, leaving Rachel's books on the table.

Other possibilities include

He walked out; Rachel's books were left on the table.

or perhaps

He walked out, with Rachel's books (being) left on the table.

Solution 3:

Not to put too fine a point on it, but they're both rather awkward. I'd recast them this way:

He walked out, leaving his books on the table.

EDIT

Given your new information, I think you are introducing unnecessary ambiguity with the two meanings of left. It's not clear who is doing the leaving of books, or if nobody is, as left can be used with or without an object. After your edits I would recast the sentence a different way:

He walked out; Rachel's books remained on the table.

or

He walked out; Rachel's books stayed on the table.

Solution 4:

Why make it so complicated? Write:

He left Rachel's books on the table and walked out.

Solution 5:

Both examples are absolute constructions. They are perfectly correct, but in most cases rather literary and old fashioned. I think Jason's explanation of what difference being makes is right, though I find it hard to explain why being makes a difference here.

The absolute construction is like a regular, attributive participial phrase, except that the agent/subject of the participle is not a noun or pronoun from the main clause but is itself a part of the construction:

Provided that she receive no visitors, she will be allowed to stay at home. (absolute construction with past participle)

Her mother dead, there was nothing keeping her here. (w. adjective)

Her father dying, she felt she could not leave the island. (w. present participle)

In attributive phrases, the participle or adjective is an attribute to a head noun or pronoun outside the participial phrase (attribute in bold, head in italics):

Roughly measured, the height of the table seemed about right.

Hit hard by the earthquake, Japan will need time to mourn its dead.

Still wet behind the ears, he assumed she would indeed show him around her bedroom.

Giving her the ring, he spoke: et nunc moriere!

Both attributive participial phrases and absolute constructions can be anywhere within a sentence:

There was nothing she could do, her husband being such a jerk. (abs.)

The King, his crown (having been) destroyed by the assassin, had lost his last shred of dignity. (abs.)

Nurse Ursula, suffering from a terrible disease, still managed to cure all patients with her magic touch. (attributive)

Nurse Ursula cured all patients suffering from pneumonia with her magic touch. (attributive)

Usually the head of an attributive participle is the subject of the main clause; but not always, as you can see in my second Nurse Ursula sentence.