When you 连忙 do something, you do it swiftly but calmly and naturally, you're not lack of time. And when you 急忙 do something, you do it in a hurry and rush, while you can hardly finish it in time.

1.In terms of meaning,They both shows speediness of actions and the usage are basically similar.But 急忙 emphasize the haste in the mind,and stress on describing the expression on face. 连忙emphasize the haste on the time of the action, and often stress on indicating some action immediately happens after the last action in the time . e.g. 上课时间快到了，他急忙穿衣服出门。（It's time to class,and he put on hurriedly and go away from home） e.g.他不小心撞了一个女孩，连忙向她道歉。（He apologized to a girl fleetly whom he just bumped into . ） 2.As an adj.急忙 can be used reduplicatively.But 连忙 can not e.g.他急急忙忙的去医院看他的妻子。（He went to the hospital to see his wife quichly.）

书写不同~~ 个人认为,连忙是发生在眼前的 而急忙是还有一段时间 连忙: 　　【读音】lián máng 　　【释义】急忙，赶紧；立即，马上。 急忙: 【读音】jí máng 【释义】急速匆忙；赶紧。

basically they are very similar usually we don't tell them apart

连忙 focuses on stressing the compact continuity between two different things,(连：continuous) 急忙 focuses on stressing the urgency. (急：hurried) For example, 看见有老人上车，年轻人连忙起来让座。 约会要迟到了，我急忙跑着过去。