They are both used for singular 'you', they are both used for Accusative and Genitive.
"cię" is the neutral form, the basic one. "ciebie" is emphatic.
so "Lubię cię" is a normal sentence meaning "I like you". You could say "Lubię ciebie", but it sounds like "I like you (but not him)".
The emphatic form is a must at the beginning of the sentence (granted, such sentences aren't common). So "Ciebie lubię" is like "You - I like". I can imagine a dialogue "-You don't like anyone! - I like you." and "Ciebie lubię" could work there. But as I said, it's not common.
The most important thing is that prepositions need the emphatic form (for pronouns that have such a distinction, of course). so "Jestem blisko cię" wouldn't just be unnatural, it would simply be wrong.
Actually some pronouns have a special n- form that is used only after a preposition and it's a must then. For example let's take Genitive masculine: "Nienawidzę go" (I hate him), "Kocham ją, a jego nienawidzę" (I love her, and/but him - I hate), "Idę do niego" (I am going to him). All instances of 'him' are in Genitive and they have different forms.
Do you mean:
Kocham ją, a jego nienawidzę.
There's a contrast in this sentence, amplified by the contrastive conjunction "a" = but/whereas. Hence the emphatic form of "go" is necessary here.
I love her, but HIM I hate.
Such a word order in English is rare, but I believe it illustrates well how "him" is emphatic.