What exactly is different about the slice of space and lightcone concept in special relativity?

In Newtonian physics it is assumed that ‘now’ is the same moment in time everywhere. The constancy of the speed of light conflicts with that, so in SR there isn’t a common ‘now’. However, since everyone agrees that the speed of light is constant, you can use that fact to set up a standard way of reconciling times and distances across different reference frames.

In Newtonian physics the fact of a common ‘now’ allowed for events to be given a definite order in time. In SR that is not the case generally. What you can do is say that events inside a light cone definitely happen at a later time than the light cone’s origin, and that’s one reason why light cones have an important role in the absence of a common now everywhere.

The key difference between "light cones" in relativity and "now" in Newtonian physics is that they define conceptually different things. The relativistic "light cones" define causality, the set of events that can cause a given event and the set of events that can be caused by a given event. It distinguishes cause from effect. The Newtonian "now" defines simultaneity, the set of events that are before and the set of events that are after a given event. It distinguishes past from future.

In relativity, nature doesn't care about simultaneity, it cares about causality. Simultaneity is a purely human or mathematical convention.

An event is a place and time. A spherical pulse of light emitted from that event traces out the light cone of that event. It is all the events that the pulse will pass through. In a year, the pulse will pass through a shell with a radius of a light year. In 2 years, it will pass through a bigger shell. Etc.

Space time is all of space progressing from past to future. It includes all points of space at all times. A slice of space time is all of space at some fixed time.

It is a useful idea if you want to measure the length of a moving stick. You hold up a rule that is at rest with respect to you. As the stick passes by, you record the positions of each end. It is of course important to record both ends at the same time.

In Newtonian physics, there is no problem. Time is absolute. Everybody, moving or not, agrees which events are at the same time.

This is not true in special relativity. An observer moving with the stick would say that your observations of end positions were not simultaneous. This is why you measured a contracted length, while he measured the proper length of the stick. Of course, you both correctly measured the length in your reference frame.

A light cone is very helpful for determining when two events are simultaneous, and therefore which event belong to the same slice. For your stick measurement, you put a light source at the midpoint of the stick. Everybody agrees that the speed of light is constant, and therefore light will reach the end at the same time. And it does.

But again, you and the moving observer get different results. You identify two events where the light passes the ends of the stick. These are simultaneous to you. The moving observer says you measured the front end before the back end, giving you a length that is too short.

The key point is all observers agree on which events are in the light cone of an event. Which events belong to a slice of space time depends on the frame of reference.