So something that comes up fairly regularly is the question of whether Danu and Anu are the same goddess or two distinct individuals*.
I want to say up front that this is one of those fun things that scholars disagree about so what follows is not meant to be conclusive but merely reflect my opinion and the evidence I base that opinion on.
I tend to believe that Anu and Danu are different goddesses and this is why:
1. The names have different meanings. Danu is related to the word for flow while Anu has a meaning related to wealth or abundance. We also see cognates to each as distinct individuals in other Celtic cultures like the Welsh Don and Anna (although in fairness Don and Anna are also sometimes conflated)
2. Although many people try to argue that Danu is actually a confusion of Dea Anu* or De Anu there are several problems to my mind with this. Firstly it assumes that the pagan Irish and the monks writing down the stories were unfamiliar enough with their own language that they would have conflated de Anand into Danand (and then not realized this was a new deity), which I'm very doubtful about. Old Irish is not pronounced exactly as modern Irish is, for one thing, but even if it was I find it hard to believe that native speakers would have confused JAY AHN-ahn for DAHN-ahn as it requires not only dropping a vowel but also shifting and dropping a stressed syllable and changing how the initial 'd' is pronounced entirely.
3. One main example given to support the above theory is that both Danu and Anu have sites named after their paps (no really) and that this somehow suggests they were the same Goddess and the sites had the same name which over time was corrupted into two different versions. However as with the above example the problem is that in Older forms of Irish the two place names are not sufficiently similar in my opinion. Keating in his Foras Feasa ar Eirinn refers to the site attributed to Danu as "Dá Chích Dhanann" and the Onomasticon Goedelicum locorum et tribuum Hiberniae et Scotiae calls it "Dá Chich Danainne" while the Sanas Cormac refers to Anu's site as "Dā Chīch nAnund". However even if we accept that the name of this single location was incorrectly attributed to both goddesses instead of one, that doesn't prove they themselves are a single deity. Effectively the argument is that the eclpising of Danann to nDanann and of Anann to nAnann render the names identical; however this occurs only in situations where the names are eclpised, generally in the possessive. This also presupposes a sort of chicken-and-egg argument where the name of the location would have to have somehow overshadowed or obscured the deity it was named for to a degree that the local populace forgot whether it was Anu's breasts or Danu's. I find that idea highly suspect.
4. Which leads me to the main reason I disagree with the conflation of these two deities - we do have examples of each name occurring separately uneclipsed. In the Lebor Gabala Erenn we are told that Anand is the personal name of the Morrigan but the Morrigan and Danand appear together, for example, in the same redaction of that text, only a dozen verses apart:
"Ernmas had other three daughters, Badb and Macha and Morrigu, whose name was Anand"
then, shortly after in a list of women of the Tuatha de Danann:
"Danann, mother of the gods. Badb and Macha, greatness of wealth, Morrigu"
Although I know that people question the veracity of the LGE, particularly between redactions, I find it hard to credit that the authors would identify the two goddesses as different in the same version of the same text if it was understood in oral tradition that they were one being. While we can certainly criticize the written texts on several levels to give validity to the argument that Anu and Danu are the same goddess conflated in written texts we must first believe that the examples of them as individual goddess given in uneclipsed versions of their names are not reflecting genuine oral tradition, and I am just too skeptical of that to believe it. We would have to believe that people recording stories they had heard all their lives were making these mistakes, and I just can't credit that, personally. It smacks too much of the hubris of modern times saying that our ancestors were more foolish and stupid than we and unable to realize what we find obvious relating to things in their native language and with their own native culture.
I cannot say that I am right and those who believe differently are wrong, but that is what I believe and why I believe it.
* Dea Anu or De Anu meaning 'goddess Anu'. This is also problematic as goddess is usually given in Old Irish as bande not de
* The names appear in the texts as Danand and Anand and later as Danann and Anann. These are both the genetive forms, however the assumed nominatives would be Danu and Anu which are the way they often appear in modern paganism.