If the garage door opener always responds properly to the hard wired push button, the fault appears to be at the electronic board or maybe the at the relay that activates opener as soon as you push the remote's button.
The electronic board or the relay may have a faulty section / bad or loose connector / poor soldering, which while expanding and contracting during temperature and maybe humidity level changes doesn't act as it supposed to.
Once you activate the garage door opener with a hard wired push button that faulty connection might re-engage and start acting properly again (I'm just guessing).
Another thing I would suggest is to try operating your door manually. If it doesn't open from the remote control, disonnect the opener arm and try to lift the door up. You should be able to lift it easily all the way up.
Do not disconnect the opener while the garage door is in open position unless you know the its springs tension has been properly adjusted / always test on a close door!