As a career engineer officer the answer is not much difference to the officer - the officer is there to lead troops. There are two main arms to the Corps of Engineers. The "green suit", tactical side is the Engineer Regiment. In the Regiment, there are combat engineers and construction engineers. Combat engineers are more integrated with brigade combat teams and perform more tactical training as part of that combined arms team. Combat engineers in mech units have to qualify on the various weapons systems and platforms such as the Bradley as well as train and qualify on their engineer skillsets so a lot of time in the field when not deployed. Combat engineers also fight as infantry. That is a simple answer as the Regiment has changed in the past 10 years and there are Route Clearance companies, Urban Search and Rescue company(ies), Sapper companies, Bridge companies, etc. so each type unit has a primary mission set. National Guard houses most of the combat engineers in the reserve component.
Construction engineers (horizontal, vertical) are your skilled trades (plumber, electrician, carpenter, mason, equipment operator). They can be close with tactical units or elsewhere in the battlespace. Their primary function is to prepare and improve theater standard construction of bed down facilities, logistics facilities, roads, airfields, etc. Types of units are horizontal, vertical, asphalt, fire fighting, and more. There are quite a few more construction type units in the reserves vice the active component.
As an officer, you will likely move between the two types, if you choose, or if one wants, you can also stick with one or the other but should want to diversify. An officer who broadens their horizons with service in both types of units generally fair better in selection for battalion command, although that is not an absolute.
The other part of the Corps of Engineers is the civilian side or simply USACE or Corps of Engineers. This is comprised of 42 districts in the US and abroad with 35+k civilian employees and several hundred green suiters. They are charged with civil works, military construction (barracks, headquarters, hospitals, etc), navigation (dredging), regulatory (permits for wetland development and impacts to waters of the US), and other tasks. This is more in line with what some young grads think they are going to do with their civil or other engineering degree and may never have the opportunity to serve in as there are so few slots compared to the Regiment.
I had the pleasure to serve in Sapper units, combat heavy (construction), and in USACE. Happy to chat via PM if you wish.