One of those VPs left, and I was given that position. But my title is
Director, not VP.
Does this difference have any significance, especially in a privately
Yes, most likely.
In your company, since there are now two VPs and one Director, there is a clear difference, and probably a significance. For some reason, you are not quite at the same level as the other two VPs. That could be because the company wants you to earn your way into a VP title, or because you don't have the same span of authority as the other two VPs, because you aren't paid as much as the others, or for some other reason.
In general, titles mean only what the specific company wants them to mean.
Banks for example often have many, many VPs and Directors, and many levels of VPs and Directors.
I've been a Manager, Director, VP - not necessarily in that order at several companies. Often, the actual work I performed was indistinguishable.
Many times, companies will use titles to attract candidates (figuring some folks might want to work for them if they could get promoted to Director instead of Manager). Other times, companies will use titles to rank similar peers (The Senior VP is more important than the VP, who is more important than the Director, etc). Many times, companies use titles to set certain people into a higher or lower pay grade (I've seen that happen several times when a company distinguished between the titles Manager and Director, for example).