Hakata sennen gate is a new landmark in Fukuoka which built in 2014 and it is a entrance to Jotenji temple street. The gate is located in Hakata district. Hakata merged with Fukuoka to create a current Fukuoka in 19th century.
Old Fukuoka was a castle town, a seat of the Fukuoka domain which was ruled by Kuroda family during feudal Edo period. On the other hand, Hakata was a commercial town, the biggest in Kyushu. Two towns merged in Meiji era, when the feudal system was over and the name Fukuoka remained but Hakata is more widely used among Japanese. The main station of Fukuoka is Hakata station and people of Fukuoka are called Hakatakko. May be the reason is that Fukuoka is also the name of the prefecture and they would like to avoid misunderstandings.
Anyway, Hakata sennen gate and Jotenji are recommended places to visit during your stay in Fukuoka. The nearest station is Fukuoka municipal subway Kuku(Airport) line's Gion station.