If you whip out your handy document inspector (F12 in Windows), you can see that Font Awesome has applied their fonts to Bootstrap's button, which contains CSS like the following:
.btn-primary, .btn-success, .btn-warning, .btn-danger, .btn-info {
text-shadow: 0 1px 0 rgba(0,0,0,0.2);
}
This adds the text shadow of which you speak. Borders are similarly styled by Bootstrap:
.btn-danger {
color: #fff;
background-color: #d9534f;
border-color: #d43f3a;
}
Other variations may be due to differing Bootstrap versions, overriding custom CSS, etc.