$observe() is a method on the Attributes object, and as such, it can only be used to observe/watch the value change of a DOM attribute. It is only used/called inside directives. Use $observe when you need to observe/watch a DOM attribute that contains interpolation (i.e., {{}}'s).
E.g., attr1="Name: {{name}}", then in a directive: attrs.$observe('attr1', ...).
(If you try scope.$watch(attrs.attr1, ...) it won't work because of the {{}}s -- you'll get undefined.) Use $watch for everything else.
$watch() is more complicated. It can observe/watch an "expression", where the expression can be either a function or a string. If the expression is a string, it is $parse'd (i.e., evaluated as an Angular expression) into a function. (It is this function that is called every digest cycle.) The string expression can not contain {{}}'s. $watch is a method on the Scope object, so it can be used/called wherever you have access to a scope object, hence in
- a controller -- any controller -- one created via ng-view, ng-controller, or a directive controller
- a linking function in a directive, since this has access to a scope as well
Because strings are evaluated as Angular expressions, $watch is often used when you want to observe/watch a model/scope property. E.g., attr1="myModel.some_prop", then in a controller or link function: scope.$watch('myModel.some_prop', ...) or scope.$watch(attrs.attr1, ...) (or scope.$watch(attrs['attr1'], ...)).
(If you try attrs.$observe('attr1') you'll get the string myModel.some_prop, which is probably not what you want.)
As discussed in comments on @PrimosK's answer, all $observes and $watches are checked every digest cycle.
Directives with isolate scopes are more complicated. If the '@' syntax is used, you can $observe or $watch a DOM attribute that contains interpolation (i.e., {{}}'s). (The reason it works with $watch is because the '@' syntax does the interpolation for us, hence $watch sees a string without {{}}'s.) To make it easier to remember which to use when, I suggest using $observe for this case also.
To help test all of this, I wrote a Plunker that defines two directives. One (d1) does not create a new scope, the other (d2) creates an isolate scope. Each directive has the same six attributes. Each attribute is both $observe'd and $watch'ed.
<div d1 attr1="{{prop1}}-test" attr2="prop2" attr3="33" attr4="'a_string'"
attr5="a_string" attr6="{{1+aNumber}}"></div>
Look at the console log to see the differences between $observe and $watch in the linking function. Then click the link and see which $observes and $watches are triggered by the property changes made by the click handler.
Notice that when the link function runs, any attributes that contain {{}}'s are not evaluated yet (so if you try to examine the attributes, you'll get undefined). The only way to see the interpolated values is to use $observe (or $watch if using an isolate scope with '@'). Therefore, getting the values of these attributes is an asynchronous operation. (And this is why we need the $observe and $watch functions.)
Sometimes you don't need $observe or $watch. E.g., if your attribute contains a number or a boolean (not a string), just evaluate it once: attr1="22", then in, say, your linking function: var count = scope.$eval(attrs.attr1). If it is just a constant string – attr1="my string" – then just use attrs.attr1 in your directive (no need for $eval()).
See also Vojta's google group post about $watch expressions.
