They are definitely not the same.

The Bible flatly states that the Nephilim were born when "the sons of God came in unto the daughters of men". Here, we're clearly talking about sexual union, and the phrase "daughters of men" is used to indicate what we would phrase as "human women" in more modern parlance.

As to the translated "sons of God", the original Hebrew says "bene ha elohim". The first part is simply "sons of". While that word could be used to indicate literal male offspring, the understanding here should be parallel to the "daughters of". That is, it's used to indicates a "type", as in the phrase "sons of the South". So the phrase means something like "male persons of the Elohim (type)".

It's a much longer story, but the "Elohim" can essentially be considered one and the same as the "Anunnaki". There is actually a finer point to this but I am unaware of any evidence that could be used to make the distinction.

The "Anunnaki" were an expedition from the planet Nibiru, the word itself meaning "those who from heaven came to Earth". Thus, we would probably technically refer to their species as Nibiruans. Complicating this is that the Sumerians did not use the word Anunnaki themselves and always referred to the "Anuna gods", or "the gods from heaven". Here they probably were referring to "Nibiruans", at least as well as they could understand it.

The Sumerians also used the word "I.GI.GI". This word can be translated as "watchers", but I suspect this is not the sense in which they meant it. The same word has a numerical value, which is 600. There is strong evidence this was the size of the expedition, and so it seems clear that "I.GI.GI" was used to refer to the expedition members specifically.

The Bible is regrettably less specific. When writers used the Hebrew word "elohim", were they referring to the race or only to the members of the expedition? There is at present no way of distinguishing.