应该 basically means "should", while 得 děi means "must/have to/gotta". In my opinion, the real question is what is the difference between 得 děi and 必须 bìxū, both of which mean "have to". (I don't really know the answer myself.)

得is like an order and it means You force sb to do sth.but 应该is weaker than 得，it is like a suggestion and contain a Little coercive

得 usually used in spoken language , it's stronger than '' should" and more inclined to"must"， but 应该 means "should", usually usesd in a formal situation(such as in written language).

1. When you want to give advice to somebody, I think they are interchangeable or almost similar. E.g. 你应该去学校了 and 你得去学校了。 Both can de used to remind someone to go to school. But 得 might implicate a more determined attitude than 应该. 2. If you want to express your predicts or conjectures, especially in subjunctive mood, you may only use 应该 not 得. E.G. 火车应该三点就到了，怎么现在还没来？ More, 我应该早就对你死心了。 3. If you want to request or order something, you may only use 得 not 应该. E.G. 明天你必须得把货送到。 More, 你得给我好好学习！ 4. You can add 不 before 应该, but in most cases you can't add 不 before 得 E.G. 你不应该去他家 is OK. 你不得去他家 is very odd. Maybe 不得（bu de, not bu dei） is more acceptable in an ancient scholarly talking way, e.g. 不得无礼！ 不得放肆！ 不得（bu dei）is also OK if you use in a rhetorical question in which you have known the answer. E.g. 都四点了，你不得回家了吗？

We usually use 应该 in a little formal situation than 得，so If you want to persuade sb 得 is better.