Basically you are asking why you cannot see yourself stop moving through time. To answer that, we need to distinguish between time and flow of time.
To distinguish between events happening at the same point in space but at different times we need to specify when an event happened as well as where it happened, so we add a time coordinate t. Events can then be uniquely located by their spacetime coordinates (t,x,y,z). To a physicist time is just a coordinate used to specify events in spacetime.
So spatial coordinates are observer dependent. However time is absolute. Assuming we both use Greenwich Mean Time (or some other similar standard) we will always both agree on the time no matter where we are on Earth or however we are moving relative to each other. In Newtonian mechanics time is special for this reason, so it makes sense to consider it separately from space.
The point of all this is that in relativity time is not uniquely defined. When we consider the coordinates used by different observers we find that time and space get mixed up with each other. Time is no longer distinct from space, and that’s why physicists treat it as just one of the four coordinates that together make up four dimensional spacetime.
Now why does time flow (in the everyday sense)? The answer is entropy. Entropy is an extensive property of a thermodynamic system. It is related to the microstates that a macroscopic system can have, and is consistent with the macroscopic quantities a system can have. The second law of thermodynamics states that the entropy of an isolated system never decreases over time. Isolated systems spontaneously evolve towards thermodynamic equilibrium, maximum entropy.
Our scientific time definition uses the concept of entropy to codify change in space, and entropy tells us that there exists an arrow of time.
In special relativity and general relativity time is defined as a fourth coordinate on par with the three space directions, with an extension to imaginary numbers for the mathematical transformations involved. The successful description of nature, particularly by special relativity, confirms the use of time as a coordinate on par with the space coordinates.
It is the arrow of time that distinguishes it in behavior from the other coordinates as far as the theoretical description of nature goes.
Now basically in our own reference frame, we cannot stop the flow of time, because we do have rest mass. It does not make sense to say things like what would we see from the reference frame of a photon (massless). Because massless particles do not have a reference frame. Sometimes you hear phrases like photons do not experience time or photons see time stop. In reality, photons do not have a proper time. And this has to do with them being massless. As long as you are in a reference frame of an object that does have rest mass, you cannot stop the flow of time.
How to define the proper time of a photon?
So basically massless particles are the only ones that do not have a proper time, but since your question is about objects (you do not specify it, but I assume with rest mass), you cannot stop the flow of time in your frame. To do that (to experience time differently) you would have to lose rest mass.
You are basically asking what makes the temporal dimension different from the spatial ones, and the answer is the arrow of time.