As you asking I am telling major deference between both bank is United bank of India is a public bank while Axis Bank is a private sector bank

United Bank of India is an Indian government-owned financial services company headquartered in Kolkata, West Bengal, India.

Bank profile :

Type Public
Traded as NSE: UNITEDBNK
BSE: 533171
ISIN INE695A01019
Industry Banking
Financial services
Founded 1950; 66 years ago
Headquarters Kolkata, West Bengal, India
Key people P. Srinivas (MD & CEO), Shri K.V.Rama Murthy (ED), Shri Sanjay Arya (ED)
Products Finance and Insurance
Consumer Banking
Corporate Banking
Investment Banking
Investment Management
Private Equity
Mortgages
Revenue Rs. 10,318.06 crore (2013)
Rs. 8,693.99 crore (2012)
Net income Rs. 391.90 crore (2013)
Rs. 632.53 crore (2012)
Total assets Rs. 114,615.10 crore (2013)
Rs. 102,010.39 crore (2012)
Total equity Rs. 5,883.72 crore (2013)
Rs. 5,579.69 crore (2012)
Owner Government of India

Axis Bank Limited is the third largest private sector bank in India.
Bank profile :

Type Private
Traded as BSE: 532215
LSE: AXBC
NSE: AXISBANK
Industry Banking, Financial services
Founded 1993 (as UTI Bank)
Headquarters Mumbai, Maharashtra, India
Key people Shikha Sharma (MD & CEO) Sanjiv Misra (Chairman)
Products Credit cards, consumer banking, corporate banking, finance and insurance, investment banking, mortgage loans, private banking, private equity, wealth management
Revenue 41,409.25 crore (US$6.2 billion) (2016)
Net income 8,357.59 crore (US$1.2 billion) (2016)
Total assets 525,468 crore (US$78 billion) (2016)
Total equity 476.57 crore (US$71 million)
Number of employees 42,420 (March 2014)