The Two Roles of "That"
The word "that" plays two roles in English.
Sometimes it functions as a demonstrative, occurring before a noun, as in:
- That dog is mean.
- That waiter insulted me.
When it appears in this role, it cannot be dropped. This is shown by the ill-formedness of the following:
- *Dog is mean.
- *Waiter insulted me.
The other role "that" plays is as a complementizer, occurring before a clause, as in:
- I believe that all dogs are mammals.
- She thinks that apples are tasty.
- That John went to college means a lot to her.
When complementizer "that" appears in roles like (5) and (6), i.e. embedded beneath a verb like "believe" and "thinks", it is fine to drop, as in:
- I believe all dogs are mammals.
- She thinks apples are tasty.
When complementizer "that" appears in roles like (7), i.e. not embedded beneath a verb, you cannot drop it:
- *John went to college means a lot to her.
See Radford's Transformational Grammar: A First Course (1988) for more about this dropping.
How can you tell which role "that" is playing?
If what appears after "that" could stand alone as a sentence, and the whole "that"-construction is appearing after a verb like "thinks" or "believes", then "that" is functioning as a complementizer and can be dropped.
For example, in (8), "all dogs are mammals" (appearing after "believes") could stand alone as a full sentence. Similarly, in (9), "apples are tasty" (appearing after "thinks") could stand alone as a full sentence.
What "that" is doing when it acts as a complementizer is transforming a clause into something that can be the subject or object of a sentence. Verbs like "think" and "believe", i.e. verbs which concern so-called propositional attitudes, regularly take clausal objects, so you'll often see complementizer "that" appearing (or being dropped) after these verbs.
Verbs like "notice" and "note" (which you use in your example) operate in the same way.
In short, when "that" is acting as a complementizer of a clausal object, it is okay to drop it.
Why are we allowed to drop the complementizer?
Why exactly we are allowed to drop the complementizer is a mystery, as far as I know. It's just a rule of our syntax. If I had to speculate, I would say that we are allowed to drop the complementizer because the verbs that it would occur with are so strongly associated with clausal complements that there is no need for an overt indication of complementization.
In other languages there are different lexical items for the two functions of "that". For example, in French and Italian, the complementizer function is achieved with que and che respectively, whereas the demonstrative function is achieved with ce/cette and quel/quella respectively.