Q:

Why "abdómen", "António", and "assintota" in Portugal but "abdômen", "Antônio", and "assimptota" in Brazil?

As in, why does abdomen have to be abdómen or abdômen?
I mean, who defined this difference? Who says that theses words need to be spelled differently?  
Why do the Portuguese write António and Brazilians Antônio? Are both pronounced the same way? 
Same thing for ómega and ômega? Was the Portuguese Language Orthographic Agreement responsible for all these differences, or did they come about for cultural needs and urgency? Another thing is assimptota and assintota! Why in the heck would someone create or demand such insignificant alterations?

A:

Those differences just reflect differences in pronunciation: that p in assimptota/assintota is pronounced in Brazil but not in Portugal; and that o is pronounced open in Portugal as the o, both in Brazil and Portugal, in roda, pólen, pódio, or gótico, whereas in Brazil it is closed, as in como, ovo or trôpego.
Generally nobody sets out to create these differences. It’s just that pronunciation keeps evolving, and it keeps evolving differently in different places. Now, this used to be no matter. Until the early 20th century these words use to be spelled the same in both Portugal and Brazil, namely abdomen, Antonio, omega, and, brace yourself, asymptota.
What happened was that first Portugal in 1911 and then Brazil in 1943 decided to use accents to show the stressed syllable, namely an acute one if the sound was open, and a circumflex one when the sound was closed. In most cases the sound is either open (lógica, trópico) or closed (trôpego, cômputo) in both in Portugal and Brazil. But some words have a closed sound in Brazil where in Portugal it is open, and so different spellings were born. I think this is the case of all words with a stressed o at the end of a syllable when the next syllable starts with an m or n (e.g. cômico, tônica, errôneo in Brazil but cómico, tónica, erróneoin Portugal). When we have a stressed e instead of o, like cénico/cênico, the pattern is the same, except that there are a few words that take the circumflex accent in Portugal just as they do in Brazil, like fêmea and sêmola. If we have an a instead it’s always a circumflex in both countries: câmara, tâmara, Tânia.
With words like assimptota/assintota (or their variants assímptota/assíntota) it’s slightly different. Portuguese language used to be spelled with lots of unpronounced consonants just because those consonant existed in Latin or Greek. For instance we used to spell escriptor, even though nobody pronounced the p, just because it comes from Latim scriptor. Again, Portugal first in 1911 and then Brazil in 1943 abolished these silent consonants. What happened was that some consonants were still pronounced in one country but no longer in the other, so more different spellings were born. However assímptota/assimptota only lost its p in Portugal with the 1990 Acordo Ortográfico (AO90). So I suppose the p was still pronounced in Portugal some decades ago, in the meantime ceased to do so, and they updated the spelling with the AO90 .

You can read the rules governing accents in the Acordo Ortográfico itself. The rule pertaining to abdómen/abdômen is in Base IX, number 2.a) and 5.a):

2 Recebem, no entanto, acento agudo:
a) As palavras paroxítonas que apresentam na sílaba tónica/tônica as vogais abertas grafadas a, e, o e ainda i ou u e que terminam em -l, -n, -r, -x e -ps, assim como, salvo raras exceções, as respectivas formas do plural [...] dócil [...] dólmen
[...]
5 Recebem acento circunflexo:
a) As palavras paroxítonas que contêm na sílaba tónica/tônica as vogais fechadas com a grafia a, e, o e que terminam em -l, -n, -r, ou -x, assim como as respetivas formas do plural, algumas das quais se tornam proparoxítonas: cônsul (pl. cônsules) [...]

The rules pertaining to António and ómega are in Base XI, number 3:

3  Levam acento agudo ou acento circunflexo as palavras proparoxítonas, reais ou aparentes, cujas vogais tónicas/tônicas grafadas e ou o estão em final de sílaba e são seguidas das consoantes nasais grafadas m ou n, conforme o seu timbre é, respetivamente, aberto ou fechado nas pronúncias cultas da língua: académico/acadêmico, anatómico/anatômico, cénico/cênico, cómodo/cômodo [...] Amazónia/Amazônia, António/Antônio

