Powerlifting would be better called strengthlifting. It is training for strength using a barbell. It is a sport consisting of the squat, bench press, and deadlift, though powerlifting training often includes other similar exercises such as the overhead press as well as supplementary exercises such as power variants of the Olympic lifts, isolation work, bands, chains, and even track work and yoga.
Bodybuilding is a beauty contest for men. It involves a wide variety of exercises, primarily barbells, dumbbells, and machines used for hypertrophy, as well as conditioning and strict diet protocols used for cutting weight.
There is significant overlap between the training regimes of the two, since size and strength, though different attributes, are not orthogonal. For instance, a powerlifter may use a "bodybuilding" routine during the off-season in order to build muscle size and give her body a rest from the low-rep, high-intensity work of one-rep-max powerlifting. Similarly, bodybuilders frequently use powerlifting exercises such as heavy, low-rep deadlifts, bench press, and back squats in order to promote whole-body growth and increase their strength so that they can lift greater volume with lighter weights.
When picking an approach, compare your goal to the goal of each practice. Do you want strength or to look good? It's not an entirely either-or proposition, but if health and athletics is more of a priority than looking big, it would be best to start with a powerlifting template and, if necessary, replace some parts of the program with specific elements bodybuilding exercises.
Bodybuilding has become synonymous with lifting weights in much of the world. People think of curls and the bench press when they imagine a weight room, and Arnold Schwarzennegger when they think of a lifter. This is not necessarily helpful for directing the general population to a program that fits their goals. Bodybuilding is a fine practice, but its wide array of isolation exercises performed with an emphasis on hypertrophy is not optimal for health or fitness for most people.
When picking tools, it's better to evaluate the specific exercise rather than the community or practice it comes from. Outside of bodybuilding, it's hard to justify a machine squat over a barbell squat. Outside of powerlifting, it's hard to find a reason to train strength without regard for other attributes. It's best to take what each sport does best: powerlifting's strength training, bodybuilding's conditioning and diet regimen for losing fat while retaining muscle.