Both functions are vital for practicing, and most contemporary tuners do both. Most tuners also have a calibration setting that allows you to use A440 as your tuning standard, or a range of several hertz either higher or lower. Online Beyond that, tuners come in many shapes, sizes, and prices. I use one made by Korg called the AT-1. It can sound a four octave chromatic scale, and measures seven octaves.
Some tuners use a second set of LEDs to show the tuning.
While not tiny, the AT-1 is portable enough to carry inside a music bag or case cover. I keep mine inside a small camera bag to help protect it. When you practice with a tuner you are actually working on two separate skills. First, the tuner can help you hone your ability to identify when a pitch is in or out of tune. Second, you can use a tuner to refine physical control over pitch.