Which of the two is correct?
I went to see two films today but did not like any.
I went to see two films today but did not like either.
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If you know that there is a choice between options, then you know that there are at least two options. If you know that there are at least three options, then "any" is correct.
If there might be just two options, but there also might be more than two options, then "any" is correct.
If you know that there are exactly two options, then "either" is correct.
In the original post, we know that there are exactly two movies. Thus, "either" is correct.
To my (American) ear, the following options sound even more natural:
2b. I went to see two films today, but did not like either of them.
3a. I went to see two films today, but did not like them.
3b. I went to see two movies today, but did not like them.
3c. I went to see two movies today, but did not like 'em.
"Film" is more formal than "movie".
"Any of them" and "either of them" are more formal than "them". Of course "them" is plural, and refers to "both of them" or "all of them"; "any of them" or "either of them" is singular. The "either of them" / "any of them" level of formality is needed in many legal contexts (such as laws and contracts).
"Them" is more formal than the contraction "'em". This contraction is common in spoken English, but is usually transcribed as "them" in written English.
In this case, either and any works just as well. Some answers say that any doesn't fit, and I'd like to address that:
It's the choice of words, and how alien it might feel to you, as your previous experience rails you into certain expressions.
Any and either are substitutes of each other in the example sentence(s), because of context:
Any is an inclusive words, either is "most" concrete.
Alternatively, the words you might be looking for is Neither and None:
Edit: As some might still disagree, consider that any in the first example can be considered a fragment - any [of them].
Sources about any: any ˈɛni/ determiner & pronoun determiner: any; pronoun: any
1. used to refer to one or some of a thing or number of things, no matter how much or how many.
Google translate probably: https://www.google.com/search?q=definition+of+any&ie=utf-8&oe=utf-8 Merriam-Webster: http://www.merriam-webster.com/dictionary/any
The entire Merriam-Webster definition of Any:
Full Definition of any
1 : one or some indiscriminately of whatever kind: a : one or another taken at random <ask any man you meet> b : every —used to indicate one selected without restriction <any child would know that> 2 : one, some, or all indiscriminately of whatever quantity: a : one or more —used to indicate an undetermined number or amount <have you any money> b : all —used to indicate a maximum or whole <needs any help he can get> c : a or some without reference to quantity or extent <grateful for any favor at all> 3 a : unmeasured or unlimited in amount, number, or extent <any quantity you desire> b : appreciably large or extended <could not endure it any length of time>
Any is perfectly VALID. - One or more includes 2.