'It doesn't taste that good' has two possible meanings. It probably means that it doesn't taste very nice. If the word 'that' is stressed, it means it doesn't taste as nice as someone says it tastes. "It doesn't taste as nice" means it tastes less nice than something else. It could also mean it doesn't taste as nice as it tasted the last time you tasted it. It needs context so we need what it is being compared to.

You are really asking two questions: 1. Is "that" the same as "as" 2. Is "good" the same as "nice". The answer to both questions is NO. 1. "As" makes a comparison but "that" might not make any comparison. "That" can be used to mean "very" or "so": "it doesn't taste very good". 2. "Good" is purely an indication of quality but "nice", while usually implying goodness, also implies refinement or subtlety.

Someone would understand what you meant with both- but you wouldn’t use the word nice to describe taste.