I'm not sure that подать connotes anything different, except that it means to pass something to someone, much like дать. Apparently, "passing something" always implies handing someone something, but that would be the same as дать, as far as I can tell.
So what is the difference and how can I use them each correctly, according to their meanings?
The word подать (in the sense you're talking about) means "to pass something which is close, but that you can't quite reach".
You would say подай мне, пожалуйста, соль "pass me the salt, please" if you wanted someone to pass you a salt shaker which you could otherwise get yourself, but would have to stand up and bend over the table to do that.
Or, if you were standing on a step stool and trying to fix a lamp, you would say to your wife подай мне отвёртку "hand me the screwdriver" if the screwdriver were lying on the floor next to the stool.
Подать can usually be replaced by дать, but sometimes подать is better if you need to be more precise.
For instance, this phrase:
Он стреляет, а она ему подаёт патроны.
unequivocally means that he's doing the shooting, and she's standing next to him passing him the ammunition.
If you replaced it with давать:
Он стреляет, а она ему даёт патроны
this could mean that he doesn't have his own ammo, and she provides him with it — not necessarily while he's actually shooting.
Дать is a Russian word commonly used like the English verb to give. Example:
Открой кассу и дай мне деньги. - Open the register and give me the money.
Meanwhile подать is indeed usually used as to pass or to hand. Examples:
Подай мне ключи, пожалуйста. - Hand me the keys, please.
Не мог бы ты подать мне соль? Я хочу посолить салат. - Could you pass me the salt for my salad?
Russians usually use the verb подать in a situation when they want some object which is located nearby, but are unable to grab it without getting up or walking over because the desired object is too far away from them, so they ask another person, who's near the object, to pass (подать) that object to them.
Additionally, keep in mind that the verb дать is also used as to let. Example:
Ты уже достаточно играл, дай теперь поиграть другим. - You've played enough already, let others play as well.
With the verb "давать", the emphasis is on the object that is transferred. For example, it's used when someone needs something and doesn't have it, so they ask you to give it to him. What matters is the end result — the fact that something changes position or status or ownership, but it's not very important how exactly that's to be done.
The verb "подавать" puts the focus on the mechanical motion itself. The thing you transfer somewhere isn't considered most important — what's important is who performs or controls the transferral, and how and where the object moves. For example, it's used when someone asks you to pass them some object from some inconvenient-to-reach place, or when you feed some piece of machinery you're working with, or allign it against something.
For everyday use, the difference isn't very important and you can use both words just to add one shade of meaning or other, but in some other cases — like when you talk about some aspects of engineering or machining, for example, only one of the words would be correct to use and you're expected to be precise.
