है is dropped with चाहिए in the present tense. This is probably because चाहिए sounds similar to the imperative forms of other verbs for which the copula is not used (I'm guessing here). That said, the past and future tense copulae था and होगा are used with चाहिए.
You can also drop the copula in present tense negative sentences with at least one other verb. This is a matter of preference though and many people don't drop it.
When used with nouns (like पानी in this sentence), both चाहिए and the other forms of चाहना (like चाहती) mean 'want'. But चाहिए conveys a stronger desire (verging on need) or a more immediate desire.
So, नेहा पानी चाहती हूँ can mean that Neha wants water sometime in the future whereas नेहा को पानी चाहिए means that she wants water right now.
When used with verbs, other forms of चाहना still mean 'want' but चाहिए means 'should'/'need to'.
Eg: नेहा पानी पीना चाहती है - Neha wants to drink water
नेहा को पानी पीना चाहिए - Neha should drink water