The flat tappet (or lifter) was simpler to manufacture...and the cam required careful break in if you installed a new cam. The lifter "foot", that rode against the cam lobe, rotated slightly as it went up and down. The lifter would break in with that particular lobe, so if you ever pulled the cam or lifters, you better have them labeled so that they go back to the paired set. For most V-8s, a very high moly content lube was put on new cams, that then had to be run 20-30 minutes at 2,000 - 2,500 RPM right after start to establish that lifter rotation and break it in right, otherwise, you risked abnormal cam wear and poor performance. The higher the lift (and valve spring pressure) the more critical this step was....
There were some very high performance engines built with flat tappet engines. Flat tappets fell into two categories: hydrauic and mechanical. The hydraulic lifters used oil pressure to "pump" them up to establish the valve train clearance. This was the most common and required no maintenance, as the valve train wore, the clearance was maintained by the lifter itself. For high profile cams, like the early 60's fuelie V-8s from Chevy for example, or the 327/365 in the '64 - '65 Vette, the hydraulic lifter would not work well under the high lift of the cam, so they went to a "mechanical" lifter, which was really just a metal foot under the pushrod. The clearance had to be adjusted on these periodically to keep the valvetrain right...and they would clatter a bit until the engine was warm...
Just to complicate (or clarify) the discussion a bit more, while OHV engines (think American V-8) used to be flat tappets - many are now rollers...
However, the 1932 Packard 8, which is a flathead (valves in the block, adjacent to the cylinders) has roller tappets on the cam...
So, you can't always tell by engine configuration or age of the car whether it's flat tappet or roller...
Roller or not, the Packard gets an HDEO for a variety of reasons, ZDDP among them...