Question: It seems as if both words mean *to humiliate and degrade*?

Answer: 
I wonder if you're conflating or confusing words here…

* abase - behave in a way so as to belittle or degrade
* abash - embarrassed, disconcerted, or ashamed
* bash - (3) criticize severely
* debase - reduce (something) in quality or value; degrade

From the note on **humble** in the *New Oxford American Dictionary*:

> 
> *Humble* and *humiliate* sound similar, but *humiliate* emphasizes shame and the loss of self-respect and usually takes place in public (: *humiliated by her tearful outburst*), while *humble* is a milder term implying a lowering of one's pride or rank (: *to humble the arrogant professor by pointing out his mistake*).
> 
> 
> ***Abase*** suggests groveling or a sense of inferiority and is usually used reflexively (: *got down on his knees and abased himself before the king*), while *demean* is more likely to imply a loss of dignity or social standing (: *refused to demean herself by marrying a common laborer*).
> 
> 
> When used to describe things, *debase* means a deterioration in the quality or value of something (: *a currency debased by the country's political turmoil*), but in reference to people it connotes a weakening of moral standards or character (: *debased himself by accepting bribes*).
> 
> 
> *Degrade* is even stronger, suggesting the destruction of a person's character through degenerate or shameful behavior (: *degraded by long association with criminals*).
> 
> 
>
The main difference is that 'abase' is something you do yourself (or other people do themselves, of their own volition), whereas 'abash' is something that you do to other people (or other people do to you).

I'm also not convinced that 'abase' really means 'humiliate' or 'degrade', unless you assume that any signs of abasement are automatically humiliating and degrading. That was certainly not the original meaning of 'abase'; I'm not sure it is yet an acquired meaning.

---

From an ancient Chambers Twentieth Century Dictionary (1978 reprint of 1972 edition with 1977 supplement), page 1:

**abase**, *v.t.* to lower: to cast down: to humble: to degrade. -- *adj.* **abased**, lowered. -- *n.* **abasement**. [O.Fr. *abaisser*, to bring low--L. *ad*, to, L.L. *bassus*, low.]

**abash**, *v.t.* to strike with shame; to put out of countenance: to astound: to confound.-- *adjs*. **abashed**: **abashless**, shameless: unabashed.-- *n.* **abashment**. [O.Fr. *esbahir* -- pfx. *es-* (L. *ex*, out), *bahir*, to astound--interj. *bah*.]

---

It seems clear to me that you do not normally abash yourself, so someone else does it to you.

It is not so clear that you do normally abase yourself, but the verb is often used reflexively - 'he abased himself'. I'm not sure how you could abase someone else.