Name of Hakata is very famous for japanese people except people who live in Fukuoka prefecture.
I mean that there is the area named as Hakata, but it is actually one part of Fukuoka city at this time.
But I think "Hakata" is more popular than "Fukuoka" for Japanese people.
For example there is a Paddington in London, if British people called "paddington" instead of expression of London.

From wikitravel

Fukuoka is a modern city; most of its buildings are new. Historically, it was divided into Hakata (博多) and Fukuoka, two separate cities divided by the central river, and the main railway station and port are still known as Hakata Station and Hakata Port. The westernmost part of Fukuoka is also known as Sawara (早良).

Fukuoka is a good starting point to begin a vacation to Japan for first-timers. Compared to Tokyo, even New York, London, Paris, and Los Angeles will seem sleepy and life appear slow. Beginning the trip at Fukuoka will help ease the transition. While still a big modern city, it's not hard to get around. The train station is already the loci of one of the city's two city centers (the other being Tenjin). When you're ready to head to the big cities up north, just board the "shinkansen" train line.

Yatai, "street stalls", in Hakata are now famous all over the country. Today, there are more than two hundred stalls in Fukuoka City and most of them stand close along the streets of the central parts of the city such as Tenjin and Nakasu. Yatai have become firmly fixed as a natural feature in the evening of Hakata.
Why are they so popular? There are many reasons such as "cheap and delicious", "abundant dishes", and "people to people contact". "Furthermore, that it has many trend-oriented young female customers is the recent feature of Yatai in Hakata."
The lively Yatai in Hakata have an atmosphere common to cities in Asian countries. This may be why the cultural theory in which the closest city to Asia is Fukuoka City tends to emerge.