In general, monocots are simpler in structure than dicots. Monocots evolved from dicots.
The simplest difference, and one that seems almost trivial, is the number of cotyledons[?] that these flowers have; monocots, as their name suggests, have one seed leaf. Dicots have two.
In monocot stems[?], the vascular tissue, the phloem and xylem are in bundles scattered throughout the stem, and they usually lack a vascular cambium. In dicots, the phloem and xylem are in rings around each other. They nearly always have cambium. Do not fall into the trap of thinking that a pine is a dicot because of its stem structure. It is a conifer, which is not a flowering plant. Obviously, it couldn't be monocot or a dicot.
Partly as a consequence of the above, in monocots, there is very little new phloem and xylem added to the stem. Thus, monocot stems do not grow significantly thicker each year. Any change in thickness is due to the cells getting very slightly bigger. This is why there are very few monocot trees (palms being an important exception). On the other hand, dicot stems can add new vascular tissue and thus grow thicker yearly. Most flowering trees are dicots.
In monocot roots, the phloem and xylem alternate like the spokes of a wheel. In dicot roots, there is a single, X-shaped mass of xylem at the centre, with phloem between the arms of the X.
Throughout the whole plant, monocots have more vascular tissue than dicots of similar size.
Monocot pollen has a single pore on the outer layer. This means it is monosulcate. Dicot pollen is triporate. In other words, it has three pores on the stem.
Lastly, monocot flowers (even grass has flowers, though they are not showy because they are wind pollinated) have flower parts in multiples of three. Dicot flower parts are in multiples of four or five.