First, there is nothing wrong with your idea. It will make a quite capable supply if you use it within its limitations. You can add an inexpensive panel meter to show you the output voltage (get the three-wire type that has a separate supply line).
In terms of limitations- your LM317 (+ heatsink) supply will only be adjustable from 1.25V to about 17V without adding more circuitry. In practice this is a very useful range. There will be no current limit other than the limiting inherent in the LM317 and the brick (probably around 1.5A to 2A). Bench supplies often have an adjustable current limit which is useful when you are gingerly powering something up for the first time, for example.
Ripple may be worse than a linear supply, especially at high frequencies, depending on whether you try to filter it well or not.
Power dissipation limiting will be by overtemperature cutout on the LM317 chip, which is pretty rough on it (it limits way above the absolute maximum listed temperature for operation), so you may have to be a bit more careful.
Many bench supplies are only optionally earthed, and have a jumper to allow earthing one side of the output. Your brick is probably permanently earthed on the minus side, so you cannot put supplies in series, for example. It would make damaging things that are already grounded such as USB ports more likely. Again, you can be careful.