What are the main differences (besides the wording of the Omer-counting itself) between the Sepharadic and Ashkenazi customs during Omer?
A large proportion of Ashenazim use a mourning period from the first day of Rosh Chodesh Iyar until the 3rd of Sivan (although Lag B'Omer itself is a "window", a one-day respite from mourning).
This is in memory of the Crusades that happened during this period and in general affected Ashkenazi communities.
Ashkenazim therefore also recite "Av Harachamim" on the Shabbat immediately before Rosh Chodesh Sivan and the one between Rosh Chodesh Sivan and Shavuot (if there is one) even though one would not normally recite it on those days due to M'varchin Hachodesh and a period in which Tachanun is not recited.
It is not universal but as far as I'm aware, using this as the period of mourning is not prominent among Sefardim who use the more "traditional" period from Pesach until the Lag B'Omer (many include Lag B'Omer itself and lift the restrictions the day after)
Here is a Chabad source: