Answer:
Interested - You know someone, physically they are attractive enough that you would consider them as a dating partner, what little you know of their personality appeals to you and as a result, in your mind, you have nominated them as a candidate for dating.
Like - You know someone pretty well, they are attractive enough to date, and your two personalities seem to click. There is genuine affection between the two of you. Either you are dating, or you definitely want to date that person.