Sensitive is a more common word used to refer to other being aware of their own and other people's feelings. Spiky is a slang term which mean having a somewhat prickly or edgy personality. The 2 do not mean the same.

Sensitive means to be more aware of certain things. A person may be sensitive to smells, able to detect them more than others. Spiky means overly-sensitive. If you ask a person how old they are and they get very angry over it, that would be spiky.