They are the same in meaning. Originally there was "have" and "have to". But "have" started to be used with "got", for emphasis and to help with rhythm. ("I have time" => "I have got time" => "I've got time"), This popular "have got" was often abbreviated to just "got" ("I got time"), especially in speech. So "I have", "I have got", "I've got", "I got" are all commonly used with no difference in meaning. And "got" in such usages has nothing to do with the past sense - it's just another way of saying "have". In fact in the US they even say things like "Do you got time?" which might look incorrect but just means "Do you have time". All this applies to "have to" as well. So you can use "I have to", "I've got to", or "I got to" interchangeably. And in speech, "got to" is heard as "gotta", and some people even spell it like that. "got to" and "gotta" is popular in speech because it sounds more pronounced and rhythmical than "have to". But the proper (conservative) way to write it is "have to" and "have got to".

Gotta is just the short form or the casual pronunciation of the verb got to. Both have to and got to are very similar in meaning but they definitely have different connotations. Have to has a negative connotation because it is used more for things which are obligatory but get to is more positive and it is used more in case of appreciation or gratitude.

The meaning is the same. It's just that 'gotta' isn't considered to be 'proper' English. http://www.oxfordlearnersdictionaries.com/definition/english/gotta?q=gotta