To Start / To Start On

It means pretty much the same thing. :)

However, in other senses, it could mean different things. For example:

Have you started the project?
Have you started on the project?

The first one could mean that you prepared the project but you just need to do something to start it.
The second one could mean that you haven't done any of the preparations/project at all, and you need to start on that.

:)

"Have you started your homework?" is a short version of "Have you started doing your homework?" I would always choose the latter myself.

"Have you started on your homework?" is equivalent to "Have you made a start on (doing) your homework?" It is not significantly different, but I think it might be more likely to be used when the homework is expected to take a long time to do. If so, it fits a bit less well with the following sentence, but I'm not really sure about this. All I know is that when I first read it, it looked awkward to me, whereas "Have you started on your term project" looks a bit better to me (a term project presumably takes longer to finish).

In your context, the meaning is basically the same.

In general, however, when you "start" or "start on" a task, that means you start working on it; but when you "start" a machine, that means you cause it to start doing whatever it is that it does.

"Have you started the lawn mower yet?"
"Have you started on the lawn mower yet?"

The first one is asking whether you've pulled the starter cord and caused the lawn mower to start running (for the purpose of mowing the lawn, presumably). The second one is asking whether you've started doing something to the lawn mower (e.g., maybe you needed to repair it).

I'm pretty sure "to start on" is not grammatically correct but people say it anyway... I would just use "to start"