What should I choose among TOEFL and IELTS alongside GRE?  

R2A

Nikita has beautifully explained why you should take TOEFL. I share the same opinion and in addition to that, when it comes to reporting your scores to universities, it is hassle free.

I remember one of our students who had taken IELTS had real tough time to report his scores to US University. (Part of application process)

As far as differences is concerned -

1. IELTS is paper based and TOEFL is Computer based
2. Usually IELTS Exam is split in two sessions (2 days) and TOEFL can be finished in a single day.
3. Apparently, TOEFL Scoring is straight forward while IELTS is abstruse.

To put it in a nutshell, both exams check your English language proficiency in four means of communication - Listening, Reading, Writing, And Speaking. However, the way the exam is conducted, the pattern of the exam and scoring is varied.