Either the Brexit example is a simple mistake or you're confused about the mechanics behind it, so I'll elaborate to avoid people thinking it's a sound analogy:
The Brexit referendum was non-binding, so at that point nothing legally relevant had happened yet.
The UK government then decided to honour the will demonstrated by the 37.5% of the population that cast a vote in favour of leaving the EU (51.9% of the result at a 72.2% turnout). Again, this decision didn't mean anything legally relevant but the government announced its commitment, setting expectations.
At that point the UK government could have negotiated with the EU and sought alternative solutions (not that there was much left considering "do this or we'll leave" had been the negotiation strategy for quite a while).
But then on 29 March 2017 the UK government decided to start an "actual withdrawal" by triggering Article 50 of the Lisbon treaty. This set its membership to auto-destruct within two years time and was legally binding.
The problem with Article 50 is that the UK now had a mere two years until all existing treaties it was part of via the EU would simply end and nobody had bothered to prepare for negotiating any replacements yet. As of today there are still hundreds of incredibly complex treaties that haven't been sorted out yet, the most obvious problem being how to deal with Northern Ireland's chaotic land borders without upsetting the DUP (the political party that enabled the Prime Minister to stay in office after her botched early re-election).
However unlike the US and the Postal Treaty, the UK can't simply cancel its withdrawal from the EU. If it wants to revoke its invocation of Article 50, it has to get approval from every single of the EU's (other) 27 member states -- some of which don't particularly care much for the UK for various reasons.
It's like giving your boss the finger and handing in your signed three weeks notice (after using the threat of quitting during salary renegotiations for decades) and then coming back two weeks later asking them if they can please let you stay. Except you don't have one boss but 27 of them and all of them have veto powers and some of them really don't like you.
However Scotland (where a majority voted against leaving in the first place) brought forth a case that will decide whether maybe the UK could still unilaterally revoke its invocation after all. The only problem is that a ruling might take months and may come a little too late.
Even if the court rules in favour of Scotland and the UK pulls off a full 180 at the last minute, the entire ordeal would leave the UK in a worse position by having called its own bluff and shown that leaving is not actually an option they're willing to follow through on.
tl;dr: the biggest difference is that (unless the ECJ rules differently) Brexit "took effect" immediately when Article 50 was invoked, it just includes a 2 year "migration" period during which the membership status remains in effect.