Can I replace 'get started on' with 'start'?

I read this sentence, and I'm curious if it has subtle difference in meaning.

I should get started on that sooner rather than that later.

Can I just say 'I should start that on that sooner rather than that later.'?

What is the difference of get started on and start?

In the given example, yes, you can replace get started on with start.

I should start that sooner rather than that later.

However, the two are not always interchangeable. Michael Owen Sartin wrote in a comment:

There is a slight difference between 'start' and 'start on.' One can start an engine, and the engine will be running. If one starts on an engine, the implication is that he is beginning a process such as painting or repairing the engine.

Regarding your sentence though, sooner rather than later is the usual expression, without the that in between.

Get started on (something) is an idiom. -- TFD

to begin doing something; to take the first steps to do something.
"When do we get started on this project?"
"I want to get started on this right away."