What's the difference between a family medicine Dr. And an internal medicine Dr.? What's better for routine check ups?

Training. Family medicine doctors train in all ages of life and usually handle all general medical questions. They perform annual exams for women and can deliver babies. Internal medicine doctors usually don't have ob/gyn training. Internal medicine is the basis for a lot of other medical specialties like neurology, nephrology, cardiology, etc. So, it depends on your needs.
Matter of Preference. A Family Medicine Physician spends their residency training, being trained in Internal Medicine, OB/GYN, Surgery and Pediatrics. An Internal Medicine spends their entire residency in Internal Medicine, as I understand it. The Internal Medicine Doctor is trained more in depth in the Internal Medicine fields, Cardiology, Nephology, Oncology, etc.