Why is "that" unaccented in some sentences like "We knew that the next day would be difficult." and not in others?

In your first example, that is the complementiser usage, a function word used solely to affix a complement clause to the (reported speech) 'speech marker' equivalent:

• We knew [that the next day would be difficult], compare We said / thought / observed / feared / ... [that the next day would be difficult].

It's just a 'tow hook' of the correct form, like 'whether' and 'if' in similar sentences, not needing chrome plating. After certain of the verbs involved, it's often deleted (don't try this with an actual tow hook), especially in conversation:

• We knew/said/thought ... the next day would be difficult.

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It's the same with the relativiser incarnation, another function usage:

• Have you seen the house that Jack built? / Have you seen the house Jack built?

The word is unstressed, in fact 'understressed' and warranting the schwa – and again often omitted.

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However, with the demonstrative determiner

• Look at that man./!

there is a pragmatic element. 'Demonstrative' means 'pointing out clearly / crisply', and although there is an unstressed (referring to the actual identification, not the overall alerting) variant of the example sentence ('Just look at the man I'm pointing to!'), we may also choose to stress which particular man is involved:

• Look at that man! [not those you're busy watching over there].

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The same applies fot the demonstrative pronoun usage.

• Yes, we'll take a look at that.
• Will you look at that!

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You mention yet another usage of 'that', the degree / secondary / adjective modifier† (traditionally classed as adverb) usage:

• It isn’t that urgent.

Again, there is a choice as to whether to stress 'that' (here equivalent to 'so') in this usage. As a tag-on after a request for a job to be done say, the sentence may be delivered without stress to gently relieve pressure on the person so tasked. However, if the person has responded say "I'll drop everything – going to Uncle Tom's funeral, our anniversary meal, the kid's birthday parties, collecting my OBE from the Queen ...." stress is certainly warranted:

• It isn’t that urgent.

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†

The Fixed Class of Degree Words "[An] example of words that don't fit neatly into one category or another is degree words. Degree words are traditionally classified as adverbs, but actually behave [very] differently syntactically [from prototypical adverbs], always modifying adverbs or adjectives and expressing a degree: very, rather, so, too.[...] This is a relatively fixed class and new members do not enter it frequently." (Kristin Denham and Anne Lobeck, Linguistics for Everyone. Wadsworth, 2010) [Nordquist; ThoughtCo]