You need to 'admit to' something, so the second phrase is not grammatically correct. In full it would read 'I admitted to having felt some of these emotions myself', but 'admitted to some of these emotions' is accepted as a shortened form.

Thank you. Justice and Gray.

The difference may be regional. In the US, to "admit to [something]" is more correct, but simply saying you "admit [something]" is very common. Based on Adrian's answer, I suspect that perhaps in the UK, only the more correct form is common. One reason to prefer "admit to" is because to "admit" something, without "to," can also have a different meaning. If you "admit" something, then you allow it to exist or to be present. For instance, if you have a ticket to a concert, you will be "admitted" into the concert hall, meaning that you will be allowed to enter.

There’s not a difference between the meaning of these sentences. In the past, TO would be omitted, but today it is common to use TO. So, I would stick with the original sentence just because it seems more native but remember that most likely the person you are speaking to wouldn’t notice anything wrong with the second sentence.