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.