According to the Oxford Advanced Learner's Dictionary both mean the same. However, you should not say 'at anytime.' You should say 'at any time.' It's alright to say, "Call me any time." You should not say, "Call me at anytime." The question "Can I get a meal here at any time of the day?" is grammatically correct. Remember that the context is very important.
The expression 'anytime soon' is part of American English. and is used in negative sentences and questions to refer to the near future. Example: Do you think she'll be back anytime soon?