They are similar but subtly different. "Big like me" means another person has a similar height or size (weight, muscle mass, etc.). "He's a big guy, like me." He's not exactly the same size, but he's close, either a little bigger or a little smaller. Notice that you should use a comma between "big" and "like me" here. "Big as me" means that two things are identical in size or very close. "It's a fair fight. He's as big as me." He is either exactly the same size or so close that there is almost no difference. Notice that you don't use a comma here when making this comparison.

big like me and big as me means the same thing but using 'big as me' is much more common