"Take a look" and "check it out" are generally the same. However, "take a look" is considered more formal, so you are more likely to use it at work, or in a professional setting like giving a speech. "Check it out" is considered more casual, so you are more likely to use it at home, or with friends. "Take a peek" is not used much. It would be slightly different than the other two because it would refer to quickly looking at something. But again, it is not used much, if at all.

Your question should be: Is there a difference between...? (singular) or Are there any differences between....? (plural) or Is there any difference between ...? (uncountable) And the answer? Yes. 'Take a look' is neutral English, and it simply means to look at something for a short time. 'Check it out' is much more colloquial and informal, and can suggest a decision as to whether the thing being 'checked out' is good or not. 'Take a peek' means to take a very short look at something, with the implication that perhaps you shouldn't be looking at it. This phrase is mainly used when someone sees something in advance, before it is ready or complete, or before it is meant to be seen. For example, if you have made a surprise birthday cake for someone, you might say to another person before the party 'Do you want to take a quick peek at John's birthday cake?' You can also say 'Have a look' or 'Have a peek' with the same meaning.