What is the idiom for leaving the one thing because you see a new one?

The following come close.

From Wiktionary:

Proverb the grass is always greener on the other side

Other circumstances seem more desirable than one's own but in reality are often not.

Proverb a bird in the hand is worth two in the bush

It is preferable to have a small but certain advantage than a mere potential of a greater one.

Proverb Better an egg today than a hen tomorrow.

It is better to have a sure thing now than a possibility of more later.

Similar to @ELMOJO's answer ...

How about chasing the next shiny (new) object?

If you google chasing the next shiny object, you'll get many hits, most concerning Shiny Object Syndrome.

From canwilldone, for example:

Shiny Object Syndrome comes in various forms, but my variety comes in the form of dropping what I am doing now, then chasing after something new and eventually dropping that too. It leads to a feeling of doing a whole lot of starting of a whole lot of different things, but never seemingly completing anything.

This seems to fit the OP's question, taken to an extreme.

The most obvious English idiom meaning that people want the things they don't have, even though what they currently have is better, is "The grass is always greener on the other side of the fence.

Why would that not fill your bill?