The difference can be summed up very well by one of my favorite quotes, "If you are neutral in situations of injustice, you have chosen the side of the oppressor. If an elephant has its foot on the tail of a mouse and you say that you are neutral, the mouse will not appreciate your neutrality" (Desmond Tutu). The quote essentially means that those being oppressed are hurt by noninterference. And that is exactly what "not supporting" Trump does.
I believe that by saying you "don't support" Trump, you still tolerate his behaviors and don't pay any mind to the injustice occurring around you. This is not to say that you have to necessarily "hate" Trump, but just not supporting him breeds an air of acceptance for his disastrous presidency.
A recent example would be his meeting with Putin in Helsinki. Donald Trump, by saying "Every time he sees me, he says, 'I didn't do that.' And I believe, I really believe, that when he tells me that, he means it" is committing treason. Most Americans believe that Russia meddled in the 2016 election, so Trump is actively favoring Putin over his own country. When you only "don't support" Trump, you justify his words. Simply not supporting him makes his actions acceptable.
Another example is Trump's plan to build a wall on the American-Mexican border. Not only is this against American values, but it is also a huge waste of time and resources. We cannot just "not support" such rhetoric, we must also be vehemently against it.
That is the fundamental difference between "not supporting" Trump and "hating" him. Injustice is a central theme in his presidency, and we cannot be complacent when our country and people are being affected by abuse. Together, as a country, we must not only be unsupportive of his behavior but be actively against them as well.
Sources:
https://www.vox.com/policy-and-politics/2018/7/23/...
https://www.washingtonexaminer.com/news/congress/s...