HOTP is an HMAC-based OTP, whereas TOTP is a time-based OTP. Basically, this is how each works:

• HOTP - based on a key + a counter, which increments with each code generated
• TOTP - based on a key + time, so you get a new key every N seconds

TOTP is quite common and honestly is all I use, whereas HOTP may be more common in certain enterprises. Main criticisms:

• HOTP - longer time window for a key to be valid for the entire time between logins (i.e. potentially easier to brute force)
• TOTP - less user-friendly due to the time window; also, you just need a clock, you don't need to know the counter value (if someone gets the key, they can generate keys whenever)