Both are good branches and offer a lot of opportunities after completing your graduation.

There are different perspectives in which you can compare two different departments, which obviously excludes what they deal with, because that cannot be compared.

Mechanical Engineering:

Mechanical Engineering graduates can work in the automotive, aerospace, chemical, computer, communication, paper, power generation and almost all manufacturing industries. Mechanical engineers also work in maintenance and service profiles in all industries. Also they can choose teaching profession in universities and colleges.

Mechanical Engineering have job opportunity in private sector and public sector (CSE branch doesn't give much options in public sector)

Top Recruiters for Mechanical Engineer:

BHEL, BPCL, DRDO, Indian Oil, GAIL, NTPC, Ashok Leyland, Hona, Mahindra & Mahindra, Maruti Suzuki, Tata Motors, Infosys, L&T, DMRC Etc.

Below  are job profiles in ME:

Design Engineer
Machine component designer
Technical Officer
Product Manager
Purchasing Officer
Sales & Maintenance Engineer


Computer Science Engineering:

Software engineers have been recruited by the IT majors like Tata Consultancy Services (TCS), Infosys, Wipro, HCL, Cognizant, Accenture etc. They were even recruited in the famous Silicon Valley in USA by top companies like Yahoo!, Adobe, Apple Inc, Google, Intel, etc.

CSE branch doesn't have much options in public sector.

Below are job profile in CSE:

Computer Programmer
Systems Analyst
Software Developer
Hardware Engineer
Consultant
System Engineer
System Designer
Networking Engineer
Database Administrator
Web Developer
E-commerce Specialist
Programmer

In India, the job prospect of a CS graduate is more, compared to ME, owing to the presence of a large number of IT companies and their demand for mansource. So even if you don't make it to a tier-1 college, you will land a decent job, most preferably in an IT major/MNC. Though the salaries will differ a lot from that of tier-1 colleges.
On the other hand, there ain't a lot of core companies(for ME). So, even if you are not in an tier-1 college it won't matter. You have to be exceptionally good in your curriculum, score high and develop deep understanding of the subject. As most of the top core companies recruits through GATE.