We find the words sh'eila and bakasha used a few times in the megillah: Achashveirosh asks Esther, "Mah sh'eilaseich... u'mah bakashaseich," at the first party, and she responds,"She'eilasi u'bakashashi..." is for Achashveirosh and Haman to come to the next party. At the second party Achashveirosh repeats the question, and Esther responds, "Tinasein li nafshei b'sh'eilasi v'ami b'bakashasi..."
The GR"A (I thought I had posted this once but can't find it) writes that sh'eila is a personal request; bakasha is a request on behalf of another. The Tiferes Shlomo points out that a sho'el is defined as "kol hana'ah shelo" -- when you borrow there is no cost to you and you enjoy all the benefits. Esther's sh'eila is for her own life to be spared; her bakasha is for her people to be spared.
David haMelech asks of Hashem, "Achas sha'alti me'eis Hashem osa avakesh -- shivti b'veis Hashem..." When one has the zechus to sit in the beis Hashem learning and growing it is not just to one's personal benefit -- a sh'eila -- but it is to Klal Yisrael's benefit as well, and therefore it is a bakasha as well.
The Tiferes Shlomo interprets "meshorsav sho'alim zeh la'zeh" that we say in kedusha of musaf to mean that the malachim are not serving Hashem for personal benefit. The greatest benefit for them, what they are "sho'alim," is "zeh la'zeh," to do for each other, for the next guy. Maybe that's what makes a malach -- when the thing that gives you the most pleasure is seeing someone else get something.
The Alshich and M'lo ha'Omer suggest that sh'eila is something that costs nothing for the giver; a bakasha is a greater request that has a cost. When Esther responded to Achashveirosh that "sh'eilasi u'bakashasi" is for him to come to another party, what she meant is that what is for him just a sh'eila, something of no trouble, she considers a bakasha, as if she was asking him for something great and imposing upon him, and therefore it means so much more to her.
Retuning to David haMelech's words, the M'lo ha'Omer interprets as follows: "Achas sha'alti," for you Hashem, whatever I ask is a sh'eila because there is no cost, but for us, it's "osa avakesh," the equivalent request made to another person would be a bakasha.
I didn't check how they explain the pasuk, but it makes sense to say the idea of "Mah Hashem... sho'el ki im l'yirah," means you can't lose by having yiras Shamayim -- it's only something you can gain from, not an imposition.
On the other hand, "Bakeish shalom v'rodfeihu" -- true peace is something that entails bakasha. Shalom requires compromise, and compromise means you have to be willing to give something up. Even though it's a bakasha, shalom is worth it.
See Malbi"M for yet another approach to she'ila vs bakasha.