They are the same.

Some verbs need to be followed by the infinitive 'to...'

http://www.englishpage.com/gerunds/infinitive_list.htm

Others need to be followed by the gerund '-ing'

http://www.englishpage.com/gerunds/gerund_list.htm

Some can be followed by either one, without a change in meaning - for example your case, 'like -ing' or 'like to...'

http://www.englishpage.com/gerunds/gerund_or_infinitive_same_list.htm

And others can be followed by either one, but there is a change in meaning.

http://www.englishpage.com/gerunds/gerund_or_infinitive_different_list.htm

What's the difference between them?

What's the different between them? (What's the difference between them?)

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They both mean the same thing. I removed the word "an" because it indicates a single anime rather than many.

Use "an" in a sentence like this:

"I would like to watch an anime."

"I wish I could watch an anime."