I will try, as politely as I possibly can, to explain what I think the several issues are.
1: Due to design decisions in the forum posting software there are not large blinking red buttons labeled "Clearly mark this as a WolframAlpha question", "Clearly mark this as a Mathematica question", etc. to indicate which product a poster was using and asking a question about. Changing that might be much simpler than rewriting the foundation philosophy of the notation which each product will accept.
2: WolframAlpha and Mathematica can sometimes use somewhat similar notation, but also different and incompatible notations.
3: New and novice users can sometimes not really know what notation to use to get an answer, particularly if they are asking questions about why something doesn't work.
4: More experienced users, perhaps tired of seeing a long history of questions showing simple errors in notation usage, sometimes seem to immediately assume any notation that doesn't exactly match what the product they are using expects is obviously written by someone who doesn't understand they should be using the syntax for what I assume they are using. This has happened again and again in the past here.
5: Mathematica and the notation it accepts hasn't got a quarter to take to the clue store and buy a clue to thus understand the meaning of [omega(1000) .. omega(10000)]
6: WolframAlpha and the notation it accepts understands perfectly that [omega(1000) .. omega(10000)] is meant to construct a list of the number of unique prime factors of 1000, 1001,1002, up to 10000. As a side note, I really like that you have somehow stumbled onto this method of telling WolframAlpha that you want the result of a sequence of calculations. I have tried to find a way to do similar things in the past and never succeeded.
7: If I try each of your smaller queries you provide then I see, exactly as you described, it quickly returns a list of 1001 items (on a modest speed machine using the latest version of the Chrome browser).
8: If I increase the length of the list by 1000 at a time then I see that it does take longer and longer to complete. I did see one example of "WolframAlpha doesn't understand your input.", while trying that exact query before and after did provide the correct output. But when I push it up to [omega(1000) .. omega(10000)] I see that it takes perhaps half a minute to do the calculations and a bit longer to format and display the result.
9: Thus I assume, and we unfortunately haven't any way to prove, that something crashed during the process of calculating that particular query. And we almost certainly have no way of figuring out what that was.
10: If you repeat exactly this query does it always take more than a few minutes and not finish? Or does it sometimes finish in less than a minute as mine seem to?
