提高 (tígāo): Means “to raise.” It is more general and very common. It's often used for concrete, quantifiable things: 提高价格 (raise prices), 提高速度 (increase speed), 提高分数 (raise a score), 提高水平 (raise a level/standard).
提升 (tíshēng): Means “to elevate/enhance.” It is more formal and often refers to qualitative, abstract, or holistic improvements. You 提升能力 (enhance capability), 提升形象 (elevate an image), 提升品质 (enhance quality).
Rule of Thumb: If you can measure it easily with a number (price, score, temperature), 提高 is usually the safe choice. If you are talking about an overall upgrade in quality, essence, or rank, 提升 is often better.
Incorrect: ~~我想提升我的中文考试分数。~~ (Wǒ xiǎng tíshēng wǒ de Zhōngwén kǎoshì fēnshù.)
Correct: 我想提高我的中文考试分数。 (Wǒ xiǎng tígāo wǒ de Zhōngwén kǎoshì fēnshù.)
Reason: A score is a specific number, so 提高 is the standard word.