Question: I was wondering if tension on both ends of a rope is still the same, even if there is objects with different weights on both sides of the rope.

Answer: 
I'm assuming there is a pulley involved which allows the tension from both objects to be in the same direction (i.e towards the source of gravity).

![enter image description here](https://i.stack.imgur.com/u2bzU.png)

Taking this to be the case, yes, the tension in the rope is constant. However, the entire system will accelerate, with the heavier mass moving downwards.
They can be different at certain circumstances. For example, in a pulley problem if we assume that the pulley and the rope is massless then the tension is same.[![This is the proof](https://i.stack.imgur.com/g3g5g.jpg)](https://i.stack.imgur.com/g3g5g.jpg)

But if we consider them to have considerable mass then we can see that T1 is not equal to T2.
Hope it was helpful.....