Difference between In public and In the public eye

in public—do smth. in such a way that the public may see, hear, or know:

• They will not admit it in public, but … they also know of these problems.

in the public eye—

1. (of people or things) much publicized by the media:

• His investigations on behalf of these men kept him in the public eye.

2. in the opinion of the public:

• He regretted, I think, that in the public eye he was regarded as the guy who had failed and ended up in the gutter.