促使is a neutral word,we can say 促使 sb/ sth...; but 怂恿 is a derogatory term, we can say 怂恿sb...and we cannot say 怂恿sth...

促使：褒义词 另一个是贬义词

促使：（语言、行动或者环境等）推动某物或某事使达到一定目的，这是一个中性词。 怂恿：一般指用语言鼓动别人做某事（偶尔也可指眼神等身体语言），怂恿是个贬义词。 some example for 促使 and 怂恿。 1. His evasive reply prompted me to ask another question. 他的含糊其辞的答复促使我又提了一个问题。 来自《简明英汉词典》 2. Even a small success would spur me on to greater effort. 哪怕是微小的成功也会促使我做出更大的努力。 来自《简明英汉词典》 3. This is the primary impetus behind the economic recovery. 这是促使经济复苏的主要动力。 来自《简明英汉词典》 4. The award has stimulated her into working still harder. 奖金促使她更加努力地工作。 来自《简明英汉词典》 5. Salts is a medicine that causes movements of the bowels. 泻盐是一种促使肠子运动的药物。 来自《简明英汉词典》 1. At his instigation we conceal the fact from the authorities. 我们受他的怂恿向当局隐瞒了事实。 来自《简明英汉词典》 2. His friend tempted him to steal. 他的朋友怂恿他去偷东西。 来自《简明英汉词典》 3. Who encouraged his taking drugs? 谁怂恿他吸毒的？ 来自《简明英汉词典》 4. The other boys egged him on to fight. 其他孩子怂恿他去打架。 来自《现代汉英综合大词典》 5. When my old friend Brian urged me to accept a cigarette,it was more than I could bear. 当老朋友布赖恩怂恿我接受一只香烟时，我就受不住了。 来自《用法词典》

或許你應該比較的是 唆使 (或 嗾使) 與 慫恿 吧 促使是推動使達到一定目的之意。 如：「他的一番勸勉促使她下定決心努力用功。」 慫恿是從旁勸誘或鼓動之意。 如：「他不斷從旁聳恿，老婆只好答應他。」 唆使是指使他人去做壞事。 重點在於做壞事 如：「唆使殺人者與殺人者同罪。」