Can you tell us more about what you want the clock to do?
So far, it sounds like an inexpensive microcontroller would be enough for this project. Of course, if you already know how to use the Raspberry Pi, then it would make sense to use what you already know how to use.
A Raspberry Pi would offer an advantage over a microcontroller if you wanted to communicate with the clock over ethernet or WiFi. Another advantage the R Pi would offer is the ability to display video If you wanted to use some sort of video display with your project.
How bright do you want the LEDs? Will it be used outside in the daylight? Do you want to be able to control the overall brightness of the LEDs? Do you want to be able to control the individual brightness of each LED?
Besides the brain for the clock (R Pi or microcontroller) depending on your answers to the above questions, you may want to use additional chips to assisting controlling the LEDs.
What sort of input device do you want to use to set the time? As has already been suggested, you could have the time set by using some sort of connection to your network (or internet). There's also the option of using a GPS module to set the time. There are special radio devices to receive the time.
If you don't have a way to automatically set the time each time the clock is turned on, you will probably want some sort of real time clock (RTC). RTC often come with some sort of battery or super capacitor to keep the RTC powered between uses. I've used this one
which seems to work fine. All RTC modules are not the same. Some are more accurate than others (the more accurate ones generally cost a bit more than the less accurate ones). A RTC should work fine with either the Pi or a microcontroller.
I'm a sucker for LEDs. I have a bunch of LED projects and I've made a playlist on YouTube
of a few of these projects. (None YET with the Raspberry Pi.)
Some of the individually addressable LEDs like the WS2812 (aka NeoPixels) may be difficult to control directly with the Raspberry Pi. I'm not aware of any software available which would allow the Raspberry Pi to control NeoPixels.