The traditional Upper Class were people with the greatest wealth and/or political strength. The class would often be characterised by inherited wealth or title passed down from generation to generation.
In the modern classification of class, these people might be regarding as 'elite'. However, the Upper Class were a more limited group and the ability to truly enter into the Upper Class from another class within a single generation was very difficult.
The use of Upper Class is typically related to the Royalty or those with title, perhaps landowners. As such, the economic and social capital was (is) passed from generation to generation without disruption. The cultural capital was high but in some senses was very narrow.
Because of the association with Royality and Title, the term Upper Class is often associated with the UK and 'old Europe' but this isn't really the case. The same fundamental system applies to the United States and to an extent, due to low inheritance taxation, certainly economic capital can be kept within a very small group.
Historically, the Upper Class had significant political and certainly economic control over society in general and clearly weren't keen to give any of it up. In reality, the political, economic and to an extent social capital of this class still has enourmous strength. However, the mechanism by which that power is applied is different. Historically, it was clear and obvious. In modern society, it is usually more opaque and indirect.
At the most basic level - a group which can and typically does act in their own interest at the expense of others - there is no difference. What is different is that nobody doubted that the Upper Class were acting solely in their own interest whereas the Elite are generally pretending that they are not. And in many cases, they are currently doing a good job of it.