According to govt, both are the same.
According to citizens the card you get by post is Aadhaar. The same which you download from uidai website and print and laminate is e-Aadhaar.
The difference between Aadhaar card and e-Aadhaar card:
1. Aadhaar is a unique identification proof for each and every citizen of India which enables or makes you the right person for availing all services and benefits as a member of this country. Whereas e-Aadhar card is a duplicate or printed Aadhaar card copy downloaded as a pdf file from its official website of UIDAI.
2. You will get a unique Aadhaar number with 14 digits and this number is the basic proof that you are an India. And, e- Aadhaar is also a valid PoI and can be used everywhere in place of Aadhaar card. Aadhaar cards take up to 3 months to be delivered after enrolment. So in between that time, e-Aadhar can be used.
3. Aadhar card and e-Aadhar card both are valid as PoI in every government as well as private office work.
4. It is a hard copy and it may get misplaced. Whereas e-Aadhar card is a pdf file which is secured by a password.