The tailpiece that the strings are attached to has a huge impact on the sound of an instrument.
The materials used, built in or added on fine tuners and the positioning of it in relation to the stop length is critical. Even the choice of tail gut has an influence on the sonic response of the instrument.
For example if you were to tap on a metal tailpiece you will likely be able to hear the metallic sound that rings through the instrument. Similarly, with a chunky soft wood tailpiece you are likely to find the sound dull and unresponsive. That said, if the tailpiece is a chunky softwood cheapie (often found on entry level student instruments) but attached with a piece of Kevlar gut or thin steel wire, the probability of you finding the sound and responsiveness increased will be evident.
The weight of the tailpiece is also a factor to take into consideration. If it is heavy it will slow responsiveness. So, by adding a set of four fine tuners (if the only alternative) will make the instrument noticeably slower. Additionally, it will put the strings closer to the back of the bridge therefore changing the harmonic tuning. Of course if the tail gut can be or has been shortened to compensate for this it will improve the sound. Personally, if there is limited choice because of the size of the instrument or availability, I would choose a plastic tailpiece with built in fine tuners over anything made from metal unless the quest is to create a tinny sound.
The ideal is to have a wooden tailpiece made from a select wood with built in fine tuners. The typical types of wood used are Ebony, Rosewood and Boxwood although others are also used depending on taste and available choice.