When talking about 1080p60 it delivers 60 full pictures every second whereas 1080i60 only offers halves of the pictures. Therefore the refresh rate is better when using FULL HD 1080p60 rather than 1080i60.
FULL HD 1080p delivers a picture that is smoother, cleaner and more detailed than a FULL HD 1080i picture.
TECHNICAL EXPLANATION
1080p is a frame based or progressive-scan video where you are dealing with frames. You have frame rate and it is expressed in frames per second.
1080i is a field based or interlaced video where you are dealing with fields. You have field rate and it is expressed in fields per second. A field contains half of the lines of the frame, either even lines or odd lines, and if one field is composed of even lines, then the next one will be composed of odd lines and so on.
What's the difference?
The difference between the two FHD standards lies in how the display lines are scanned. With 1080p the lines are displayed consecutively in a single pass, as opposed to the odd-numbered lines alternating with the even-numbered lines.
This means that 1080p60 pushes 60 full frames through every second whereas 1080i60 only shows 60 half frames every second, interlacing them to 30 full frames every second.