It sounds like White Fly. I use Neem Oil and it seems to work.
You just mix a couple of table spoons with water in a spray bottle and spray as needed, maybe every couple of weeks, again if needed.
Good luck!

With white fly, you will see white pupae that don't move, about 1 millimeter long, along with barely visible flyspecks.

As they mature, you will see clouds of white flies when the leaves are disturbed.

If you hit the underside of the leaves with a strong spray of water every day, that will provide decent control.

If you want to spray less often, insecticidal soap, Neem oil, or horticultural oil will do the trick.

It is probably white fly but there are other little white bugs that don't move. Do they look like the pictures on the link below?

Here is a link that might be useful: White fly image search

The more I read about the White Flys, the more I think it is not them. The tiny white spots are on the tops of the leaves only and do not move when sprayed with water. I tried shaking and spraying the leaves and they do not move. They are as so small and appear to be stuck to the leave. I cannot make out any distinctive marks because thay are so small. They also do not appear to be eating the plant but I have noticed that they are also attached to some of the branches in this vine like plant that I have. I am at a loss for what they are and how to treat them.

Neem or insecticidal soap ought to kill any bug. Maybe you've got some kind of scale.

What is the infected plant?