Differences between Verb + to be + adjectives and Verb + adjective

If you have a more illustrative title, feel free to change it. I searched but I couldn't find one.

This may be an easy and trivial question; if so, I am sorry.

What are the differences between these two sentences?

These two guys seem to be inseparable.

These two guys seem inseparable.

I can intuitively say that the latter one is grammatically wrong. Could you please explain?

EDIT: What I think is John Lawler's and FumbleFingers' answers are worth to read. The reason why I noted that someone who wants to learn the answer of this question, accidentally misses FumbleFingers' answer.

Your intuition is incorrect. They are both grammatical. And they are identical in meaning. The only difference between these two sentences is how many syllables they have.

The cluster to be, consisting of the infinitive complementizer to, plus the predicate adjective auxiliary infinitive be, is frequently deleted after the predicate seem (or appear) before a predicate adjective, like inseparable (meaning 'very close friends'). There is no specific rule saying when to perform this deletion; it's a matter of individual choice, like many other rules in English.

The reason why to be can be deleted here is that it has no meaning, and serves merely to mark the complement clause as an infinitive (required after seem) and the predicate of the complement clause as a predicate adjective (required before inseparable). So it's dispensable.

There are lots of syntactic rules (which means "processes", btw, and not rules for "Correctness" -- think of them as grammatical apps in your brain) in English that have the effect of shortening, moving, or deleting such frequently-occurring but semantically null chunks, and otherwise make speech faster. And supposedly easier.

Easier for the speaker, anyway. They don't always make things easier for the understander, or the learner, though. Frequently you have to put all that stuff back into the sentence to make it clear.

This rule (or app) is To-be-Deletion; a similar one for a different situation is Whiz-Deletion.

All verbs relevant to this construction mean something like "to be" anyway (i.e. - "something is [actually or possibly] in some state"). At the risk of crossing swords with the professionals, I disagree with John's assertion that "There is no specific rule saying when to [omit 'to be']".

(1) Verbs where we never use 'to be' involve more 'primitive' (limbic/visceral?) sensory perception as processed by the hindbrain. Anything perceived by touch/taste/smell/sound is virtually by definition "real", so you don't need to repeat the relationship with reality.

• I feel happy.
• This tastes great.
• It smells awful.
• You sound upset.

(2) Verbs where 'to be' is optional are those primarily alluding to sight/visual cortex. But of course, appearances can be deceptive - so maybe including to be more explicitly conveys that what we're seeing really is "real".

• He looks [to be] annoyed. (some may dislike 'to be' here)
• You seem [to be] clever.
• They appear [to be] friendly. (some may prefer 'to be' here)

(3) Verbs where we always use 'to be' involve far more sophisticated mental processing, such as judgement, forward planning, etc. In these cases, the state being spoken of may be only loosely connected with "reality", so we include the auxiliary verb to strengthen that relationship.

• I want to be alone.
• She hopes to be married.
• He hates to be surprised.
• They like to be active.

That looks like a pretty clear progression to me. It's all about how strongly the verb (and possibly other aspects of the context) imply that the state being spoken of corresponds to irrefutable reality as we perceive it. I'd particularly flag up the uncertainty surrounding look, where simple "sight" leans towards (1), as opposed to appear (involving evaluation), which leans towards (3).