The difference is that we vote for our representatives to parliament, on the strict basis that they fill that role for a period not exceeding 5 years. When we then have the next general election, we are not 'changing our mind'; once we vote for our MP, that result is fixed and permanent, and is not, and cannot be, over turned, because some people don't like the result / think it may be harmful.
We (the nation) voted in a free and fair referendum, which was not just authorised but instructed to occur by act of parliament. We voted to leave the EU: the exact details of what this meant were indeed unclear (other than it meant leaving the single market, leaving the common external tariff area / customs union, and ending the general jurisdiction of the ECJ, all points that were absolutely clear during the referendum). Parliament voted to enact that result, and duly started the process for leaving the EU. It would be an over-turning of democracy to stop that process: the instruction to leave the EU has been given and so should be carried out.
Once we have left the EU, then anybody is of course entirely free to make the case to then reverse that decision, just as once your duly elected MP takes his seat in parliament, you are entirely free to start campaigning, for or against him/her, for the next election.
Of course, given that the only previous time the people of the UK were consulted on membership of the EEC/EC/EU was 1975, which was a referendum won by the remain campaign, it might be viewed as hypocritical for supporters of remain to insist on another referendum in less than 40 years from 2016.