No, there isn't. In the UK, 'Have got to' and 'Have to' are used interchangeably in standard everyday speech, and in informal writing. The choice would depend on the sound and rhythm of the sentence - you'd use whichever sounded best. The 'got to' form would not be used in more formal contexts. 'Have got to' is used much less American English, where the standard form is 'Have to'. There is the very informal usage 'gotta' which is common in spoken American English (eg I gotta go!), but this is seen by most as 'incorrect'. You shouldn't write the word 'gotta', because it is a feature of colloquial spoken English.