A spinning bike is usually sturdier, to allow for standing riding at full effort.
A spinning bike will usually have a heavier flywheel, which give a smoother character to pedaling, more similar to real riding.
A spinning bike usually has a handle bar which allows for several different hand positions.
Some spinning bikes are fixed gears, meaning the pedals will keep spinning for a while when you go off power.
Spinning bikes will usually be more adjustable in saddle and bar fore/aft position.
Spinning bikes will usually be set up to take regular bicycle saddles, as opposed to the extra wide monstrosities often found on regular exercise bikes.