In media, people are often saved: taken away, by some external agent, from external turmoil. A child is pushed out of the way of a speeding truck, a princess is extracted from her abductor's castle, whatever. An alternative to this, however, is what I am calling deliverance. The best definition of this I can currently give is salvation from internal turmoil, as opposed to external (which is merely being saved). As such, it's intensely psychological.

Of course, one definition isn't enough, when it's really more of a know-it-when-you-see-it thing; to get the feeling of the difference across, I'll list some of the stylistic differences in the presentation of the two, as well as some examples of characters.

Stylistic differences between saving and deliverance:

Music: climax without modulation vs legato modulation to major key

Color palette: cold vs neutral or warm

External agent: masculine, intense determination vs feminine, calm composure

Furthermore, when someone is being saved, the audience is encouraged to identify with the savior (the doer); when someone is being delivered, the audience is encouraged to identify with the receiver. (It is important to note that the one being delivered is generally passive in this relationship).

Some examples of deliverers:

Madoka Magica, in particular, manages to showcase this relationship in a very clever way. Its main characters are 'magical girls', capable of wielding immense magical power, but this power comes from artifacts known as soul gems, of which each girl has one. When they use their power, or when they experience despair, their soul gem loses brightness; when it is filled with darkness, they die, killing many other people as well. Hence, the mechanics of this universe link internal and external turmoil, such that to deliver someone from the former is also to save them from the latter. They don't even need to make it metaphorical!

Now, for more questionable speculation: what drives the difference in depictions of the two? I can think of two factors:

First, it's just more difficult to depict deliverance. It requires characters to be fleshed out, to have real psychological states and tendencies. I'm tempted to say "latent psychological variables": states which cannot immediately be inferred, like pouty 😢 sadness or fiery 😡 anger, but which can nevertheless be inferred through subtle shifts in tone: perhaps a shift towards helplessness, despair, mania, etc. effected over a long period of time. (Evangelion is the best example of this, and it's not even close. Unfortunately, it only makes the show more miserable).

Second, a cultural aspect: to be saved is for someone to defeat external turmoil for you which you could not yourself overcome, which is no deficiency: your enemy was too strong, but you received help and survived. To be delivered is to receive salvation from internal turmoil, which is necessarily indicative of not only a deficiency, but such a deficiency that it became central to the plot and needed to be fixed by someone else. Nobody wants to be deficient, least of all deficient in such an external way, and nobody wants someone else's help for what is supposed to be "their problem". Hence, when deliverance is not shown with great skill, perhaps it comes off as pathetic and embarrassing to viewers.