Both are fine gramatically.
I agree that they are close, but to my ear:
I suggest - could either have a hint of a menacing tone (there will be consequences if you don't) or expectation to obey, coming from a speaker who is in a position of authority / expertise (boss, parent, doctor).
I would suggest - could imply a more gentle offering of advice
The really just makes the advice a little more strong. In the case of "I really suggest" coming from a boss or parent, I would definitely follow without arguing (unless you have a really great work environment where everything can be questioned).