#She came home angry

Here angry is an predicate-adjective-clause that describes the direct subject namely "She".  

#She came home angrily

Angrily is the attributive-adverb clause that attaches to the the "came home".

If you want to kind of visualize it associations are:

(she came home)(angry)


(she)(came home angrily)

If we were to use modern predicate grammar, the former would be:

    came(angry(she), home))

Whereas the later would be expressed as:

    angrily(came(she, home))

Hope that clarifies the technical difference between them, but what the heck does it mean (thank you @Kris for pointing out my silliness of not actually answering the core question - teehee)

Okay so the two expressions give an indication of different impact on the subject's behaviour. 

*She came home angry* expresses the subject's emotional state, without saying anything about its affect on the action of coming home. There is perhaps a weak cause-effect association, where her being angry may have caused her to come home. However this phrase says nothing (one way or another) about whether her being angry affected the act of coming home as she was in the state of that action.

*She came home angrily* again expresses the subject's emotional state, but it ties the state of being angry tightly with the action of coming whilst she's in the process of doing that action.  

Whatever her mode of coming home (e.g. walking, running, driving or whatever) one can almost imagine her manifesting that anger into the action of coming. For example it conjures up the image of her driving rashly or bursting the door open when she comes in. There is an expectation of outward observable channeling of that anger in the main action of coming home. 

Hopefully this is clear.  

P.S.: I noticed earlier that, if spoken left to right, ignoring parentheses, it seemed that  predicate grammar expressions sounded like Yoda speaking. It still seems so.