Wait and stay: are they the same?

It depends on who you ask.  Lots of performance people (and a few pet people) use both wait and stay, but they do NOT mean the same thing.  I use both.
 
Wait means: you wait right here, I'm going a little ways away, but I'll be telling you to do something.
Stay means: I put your butt here, don't you dare move it until I get back!
 
The dog learns that wait will always mean "I have to stay here temporarily, but in just a moment I get to do something!: and they learn that stay always means "I must remain stationary until she returns to me"

I use both basically in the way that Xeph does. For me, unlike "stay," "wait" does not necessarily mean the dog has to stay in a certain position, just that he stay where he is for a moment or two until I give him the go-ahead.

The dog learns that wait will always mean "I have to stay here temporarily, but in just a moment I get to do something!: and they learn that stay always means "I must remain stationary until she returns to me"

 
Ditto, thats basically what I would define as the difference between wait and stay.  I use both as I usually tell Dasher to "wait" on the agility line b/c I will be releasing him at some point.

In my puppy school, I know we will learn a sit-stay and a stay-come.  Is one of these an instance that I could teach a wait?  When you give your dog a wait command, are you walking away from him or are you staying with and that is the difference between a wait and a stay?
 
Sue

I use "wait" before we go out the door when Sunny is standing, or at a curb waiting to cross the street if she is not heeling. Sometimes at doors I alternate and make her sit/stay or down/stay to shake things up. Wait just means wait there in whatever position you are in.

ORIGINAL: Sue and Buddy
When you give your dog a wait command, are you walking away from him or are you staying with and that is the difference between a wait and a stay?

 
When I do a wait, I am walking away from the dog, and while the dog understands he is to remain where he is, he won't be there for more than a few seconds to a minute.  I use "wait" when Strauss is sitting at the door waiting to go out, for a recall, and during agility, before I release him to run the course.
 
Stay is for obedience only.  I use stay (well, bleib in my case, as I train in German), when I am laying a track and I need him not to move, and I use it for the long sit and long down, other than that, everything is a wait.

The dog learns that wait will always mean "I have to stay here temporarily, but in just a moment I get to do something!: and they learn that stay always means "I must remain stationary until she returns to me"

 
Exactly how I use it, too. 
 
Kate

I trained wait mainly with the food bowl, ie he had to wait until I said okay to begin eating. The only reason I don't still use "wait" at mealtimes is now he automatically waits until released. [] But that's a great situation in which to teach and use it first. I also use wait at curbs, wait when I take off his leash at the park (so he doesn't just bolt off), wait while I hide the treat in a "find it" game... it's very handy.

I use "wait" anytime I want my dog to remain still momentarily, and another command will always follow.  It's very multi-purpose for me, eg. he waits to eat (release is "ok); he waits while I walk away and will then be told to sit/down/come etc.; if he's running ahead of me in the park, "wait" will freeze him in his tracks until given another command; he waits at the kerb when needed; he waits before getting out the car; etc etc. 
 
I only use "stay" when I want my dog to remain in the position he's in (sit/down etc) without moving, until I return to him. 

You can choose!  I use "wait" as my stay word.  I think it sounds nicer.  But, you can teach your dog to stay using "stay", or even "freeze" if you want.  You can even train your dog to stay with multiple cue words (teach them one at a time, of course).

When I tell my pup "wait", I sweep my hand in front of her nose to create an invisible line, and she knows not to cross that line.  She can stand up, sit down, roll around, stand on her head, whatever she wants, so long as she waits behind that line.  "Stay" is for when she is expected to hold the same position I put her in (stand, sit or down, usually), until I release her.  For me, it's useful for her to know the difference because she runs next to my bike off leash, and I can yell "WAIT" when she gets too far ahead, and she will ause and let me catch up to her.  It's not the same command as "stay", because I haven't put her in a position.  I hope that makes sense! 

I use both as well, but more often 'wait'. Wait is a more informal command that means don't go any farther, whereas stay means stay in that position and don't move at all. I use wait before crossing roads and when waiting for other dogs to catch up. It's a command for when we're on the go and we just need to stop momentarily. Actually, I tend to say stop to tell my dog to stop, and then wait to tell her she needs to wait until I give her a release command before she goes again. It's probably not the best way to do it, because she generally doesn't stop if I say wait, and she doesn't wait if I just say stop, so I have to give her two commands when I mostly want her to do them both every time, but there you go. She just responds better with both because I tend to say stop when she's about to go somewhere I don't want her to, and sometimes that's followed by calling her over or saying "this way" rather than making her wait and then continuing on.

Anyway, I'm rambling. I hardly ever use stay. It's generally reserved for when I want her on her bed and to stay there, or as a more serious command than wait. Stay is something that means this is where I want you to remain until you've forgotten that that's where I told you to be.

I use both wait and stay. 
 
Stay - don't move until I come back to you and free you from the position. 
 
Wait - remain here until I call you to me.
 
I find using the two commands helps eliminate confusion for the dog, and increases the solidity of the stay.  It is a pet peeve of mine when an owner calls the dog to them out of a stay to practice the come command.  Since I know they are going to do it whether i approve or not so I split the commands.  One is used if you are going back to the dog.  The other is used if the dog will be called to you.