I've used both Discord's bot API as well, and can say it was way more convoluted to get up and going. Telegram's API was a joy to build in. This was in Node.js.
You need to use the "Telegram API": https://core.telegram.org/api#telegram-api (not the "Bot API") to develop your client. It uses MTProto to communicate with the server, so I suggest you just use a library that supports it like Telethon if you want to program in Python.
I've used both Discord's bot API as well, and can say it was way more convoluted to get up and going. Telegram's API was a joy to build in. This was in Node.js.