"Shooting in JPEG" just means the processing of the raw data from the sensor and conversion to the JPEG takes place inside the camera. The amount of sharpening, noise reduction, color temperature adjustment, compression, etc done inside the camera during the processing depends on the camera settings, and may or may not be controllable by the user.
RAW files contain the raw data from the sensor without any processing (with a few recent exceptions). The fundamental difference between "shooting in JPEG" and "shooting in RAW" is where the processing of sensor data to obtain the image occurs, inside the camera or on a separate computer.
When RAW files are saved and processed on a computer the user generally has more choices about the details of the processing including whether to create images in JPEG format (compressed with some amount of loss) or TIFF (not compressed with no loss). Also, if the resulting images are unsatisfactory it is possible to change the processing settings and create revised images starting with the raw sensor data.
The disadvantages of "shooting in RAW" are the files are larger and require more storage, and the processing on the computer is another step and requires time. The larger files are becoming less of an issue as larger memory cards and hard drives become less expensive.
Most recent DSLRs and more advanced compact cameras, and even a smartphone or two, have an option for having JPEGs created in the camera ("shooting in JPEG") as well as saving sensor data in a RAW file ("shooting in RAW") at the same time. This is what I do so that I immediately have JPEG images to use if needed, but can also have the RAW files for processing on a computer. The only disadvantage is fewer photos can be stored on a memory card. But with larger memory cards this may not be an issue. I'm currently using 32GB high speed SDHC memory cards (purchased for less than $40 each) and store over 1100 combined JPEG and RAW images per card.