Are you by any chance using a laptop screen? If so the reason 1080p looks fine without anti-aliasing is probably the pixel density. On my 24" display at its native resolution of 1920×1200 (i.e. 1080p with an additional 120 vertical pixels), anti-aliasing is a must on non-2D games. The pixels are about 0.3 mm to a side, or ~95 dpi, which makes them easily visible. On a 17" 1080p laptop screen however, each pixel is about 0.2 mm to a side (~130 dpi) and thus needs less AA to appear equally as "smooth" at the same viewing distance. To get a roughly equivalent pixel density (and thus equivalent "smoothness" and "sharpness") on a 24" 16:10 monitor, a resolution of about WQXGA (2560×1600) is required.
If however you are using a >=24" 1080p display at ~50 cm/1-2 ft then that may be more indicative of your visual acuity than anything else; not everyone's eyes are created equal. An individual with 20/20 vision should be able to distinguish individual pixels when they are above around 0.15 mm/side (~170dpi) at ~50cm, and be able to detect aliasing even beyond that.
With regard to the "sharpness" point, sharpness is determined by how small the smallest details are, so the higher the resolution the sharper an image will be; its that simple. Anti-aliasing has no affect on how sharp and image is, it is designed to make edges smoother (i.e. remove aliasing (jaggies), not add detail).