It's all about the isotopes they use. PET uses positron emitters (F-18, C-11,...), this positron annihilates with an electron in the body of a patient and thereby creates two gamma-rays under an angle of 180°C. The beautiful thing of PET are these two gamma-rays. When two detectors (in the ring) detect a ray at the same line, you can draw a virtual line between them (180°C!!). On the intersection of all the different "lines" lies your source of radiation. This gives PET a good sensitivity and resolution and quantification possibilities.
A SPECT isotope (Tc-99m) emits one gamma ray, so reconstruction is more difficult, only rays that fall straight on the detector are detected (otherwise you won't know from which direction it came), so the camera has to rotate to make a full image.
I hope a made it more clear to you...