As I understand it, normal Telegram channels can have members. You can join or leave a channel. Is this the same with bots? Do Telegram bots have members/subscribes in the same was as a channel or group?
No, it's not the same.
I'll explain a bit what a bot is, maybe it will help:
Basically, it's like chatting to another user, which in that case is not real and its usually a program running and doing stuff according to multiple criteria.
So it would be more correct to say a bot has users, and not members.
If you're asking from privacy perspective - the bot can save* the list of users which communicated with it.
*It will probably save the user details if it needs to save details about the conversation, maybe statistics etc.
Channels are used to broadcast messages, basically like a personal blog or a newsfeed. They can be paired with bots for additional formatting, timed posts, self-destruction, etc. I've also seen bots that did comment sections and other things before they were officially implemented.
Bots, however, can be used for pretty much anything. I've been using one to play games, some of my musician friends have apparently found a bot that discovers guitar tabs for them, etc. It's kind of like small apps inside Telegram itself. Also, while some bots are like chatting with another user, there are plenty that ask you to press buttons instead of typing in answers. I've also used a bot for channel moderation (banning, reporting spam, etc.)
Basically, channels are blogs and they can actually be enhanced and moderated by bots. Bots are software-controlled small apps that have a much wider use.