difference between "missing" and "lost"

It depends on the context. I assume you mean the dog cannot be found, it's missing. In this case, "lost dog" and "missing dog" are synonymous.

lost (adjective): if something is lost, you had it but cannot now find SYN missing

(Longman)

If you mean the dog lost its way, then you obviously cannot use the term "missing". In this case, it's a lost dog, not a missing dog.

This dog is lost. It is a lost dog.
The dog is unable to find its way home.

I have lost my dog. My dog is lost/missing.
I cannot find my dog.

There are instances where 'lost' and 'missing' mean exactly the same as here in the post. But a deeper understanding of these two terms suggest that 'lost' is subjective and somewhat psychological.

• He is lost in thoughts.

Here lost is not missing. Moreover, the subtleties of a 'lost' and 'missing' dog has been made amply clear. When you yourself is responsible for any quandary we prefer 'lost'.

In contrast, 'missing' is objective and terrestrial. It suggests what is missing may perhaps be found out(a hope lurks some where), in 'lost' it is distanced farther. The evocative power of 'missing' to rouse all the sensory feelings all at once is unique; 'lost' comes nowhere near.

• Something is missing; Laila is not home yet.

** She is all tears for missing lost love.

The order can't be reversed.

Back to the basics. In our example, a'lost dog' is exactly a 'missing dog'; no denying that.

A "missing" dog is a dog that isn't where we are looking for it. It may not be there for many reasons; it may have been stolen, killed, run away (and known to be coming back two days later), sold on, or actually lost. (Or misplaced, thanks, Kris).

If one child in a class of schoolchildren is missing, I'd hope very much that the child hasn't been lost (that's what you might say if 20 school children go on an excursion and only 19 return; one child is missing; the teacher lost one child), usually the child is ill at home, or maybe playing truant.

On the other hand, I have a box with hundreds of screws and some may be lost, but as long as I don't notice it, they are not missing.

"Missing" is also used for things that have never been there. I brought all the tools to do some repair work, but I forgot to bring a hammer. The hammer is missing; it isn't lost.

It's different talking about a dog than it would be talking about a person. You're not apt to hear "a missing dog" because the assumption is that your dog would never leave you unless it got itself lost and couldn't find its way back. On the other hand, we hear about "missing persons" because a person might run away or be abducted (a dog can be abducted, too, but it's not a likely scenario).