What does choosing one over the other do? Are there advantages and disadvantages to either for use in handgun reloading, such as 9mm?

Alliant powder- Fast powder like Bullseye is best for target loads & light bullets. The pressure will raise faster compared to a slower powder like Blue Dot. Because Blue Dot is slower to reach it maximum SAAMI pressures for the cartridge, it pushes the bullet longer, resulting in higher velocity at the same maximum pressure rated for 9mm luger. If you load a 147gr bullet, the Bullseye powders burning rate is to fast, building maximun pressure too soon, so its not listed for a 147gr bullet on Alliants load data i am looking at. Google "burn rate charts". SAAMI data is also interesting, and can be found at this link. > SAAMI INfo

Oh yeas, huge diff. Think of pressures as a curve, pressures increase as you ad dmore pwoder. Faster powders increase pressures sooner, in some powders, much sooner. Extremes ion the 9mm would be Bullseye & AA#7. With a 124grFMJ, BE reaches max pressures @ 1050fps, AA#7 @ 1180fps+. That means both are running 35Kpsi+ but you are getting higher vel w/ the slower powder. Backing off the slower powder slightly still gives you a higher vel, say 1100fps, but several 1000psi less pressure. The only downside, it takes more of the slower powder to get to that vel/prssure level. The other diadvantage to me, the faster powders are often very small charges & can easily be doubled w/o noticing, especially in larger cases. A slower powder has a larger charge & will better fill a case & duble charges often over flow.
BTW, not diff in any reloading process, shotgun or rifle.

I generally use handgun powders that fit the type of load I intend to load. Target Loads like a 38 special wadcutter get very fast powders like bullseye or 231.

Field loads, medium velocity loads and general shooting loads get medium burn rate powders such as IMR 4756 (my personal favorite for 9mm/38special and medium 357's), Universal and Unique. Magnum loads for hunting I use 2400, 4227 and 296.

Get a reloading manual anf follow its recommendations.

Another reason handgun cartridges " Generally " use faster powders are case capacity's , slower powders generally are for larger capacity cases hence a Longer burn avenue . Short cases don't handle slow powder ignition well for velocity VS bullet weight .

An extreme example would be ; A slow burn powder in say .500 S&W cartridge with 125 grain bullet ,the bullet would pop from the case before all the powder would finish burning , thus leaving the shooter with a flame trail and little velocity !.

As already stated by others ,some powders are better than others for different applications .

A lot of people think that they feel less recoil when using faster powders to achieve the same approximate muzzle velocity as compared to an appropriate charge of slower powder.

A lot of people like to ensure they have at least a 75% load density of powder for each cartridge. This does two things for them:
1- higher load density ensures more efficient burning of the powder up to a point.
2- higher load density over 50% makes it virtually impossible to double charge a case.

It is very unlikely that an appropriate charge of the faster powders will give a high load density in any but the smallest cases. Since I have been shooting handguns indooors a lot in the last few years, I have noticed that it is quite easy to load the faster powders to get a lot of flash, which is quite distracting. I load only milder, target loads and don't see as much, if any, flash using appropriate charges of slower powders. I never noticed any problem in years past shooting outdoors.