There are far more dicots than monocots.
Those who are purists about phylogeny (relationships) restrict the name Dicots to plants that evolved after the monocot/dicot split. (Just like monocots are plants that evolved after that split.) Some of the more primitive plants, like Magnolias, are, technically neither monocots nor dicots. I prefer to ignore that and use the term dicots for all non-monocot flowering plants. (Just letting you know I’m wrong to do that.)
It is thought that the earliest monocots were aquatic plants. They had little need for well organized water transport, no real need for deep taproots, apparently no need for well-developed leaf blades, certainly no need for much support since the water would hold them up. So they lost or reduced all those expensive tissues and organs. Leaves were reduced to petioles, which would become sheaths. When conditions changed, similar selection pressures led monocots to solve the same basic issues that dicots had already solved. Populations that lived in drier conditions re-evolved leaf blades and sturdier, more effective systems of transport and support. They did it differently than the dicots had always done it; parallel leaf veins, scattered vascular bundles, etc.
To this day, monocots tend to occupy wetter habitats than dicots, but that’s only a trend. There are lots of exceptions.
The competition isn’t between monocots as a whole or dicots as a whole, but between individuals and populations that happen to be monocots or dicots. What traits are most successful depend on details of habitats and ecosystems. I’m not sure you can generalize about their success at the level of these two major groups. You also have no reason to expect that all monocots will outcompete all dicots, or visa versa.
Well, in one way you can. Because of their scattered vascular bundles, monocots are very, very bad at secondary growth (growing wider). They don’t make good trees. Oh, there are palms, but palms generally don’t produce side branches and they come out of the ground as wide as they’ll ever be. They have to have very large leaves to reach sideways from the trunks and there are limits to what those leaves can accomplish in terms of claiming 3-D space. In general, monocots don’t make good shrubs, either, though Pandanus and a few others do an adequate job of it. Dicots really don’t have to worry about monocots evicting them from the woody plant niches! Restrained by their vascular system, monocots don’t even do really well in the niches for branching herbaceous plants – in general.
Monocots shine at growth forms that don’t require secondary growth. They can grow old by becoming longer rather than by growing wider. They do well as geophytes that make bulbs and send up yearly flowering shoots then move their perennial growing point to the side or up to make a new bulb. They shine is at being rhizomatous (or secondarily forming bunchgrasses by minimizing their rhizomes). These rhizomatous plants spread out under the ground and send up shoots that don’t branch much and don’t last long. Only the rhizome is long-lived. (In bunchgrasses, the “crown,” the mass of very very short rhizomes is perennial.) Think grasses, sedges, and rushes as well as rhizomatous “lilies” like Solomon’s Seal and trilliums. Very, very successful plants that don’t have the same life style as branching, leafy, often woody herbaceous plants. Grasslands and shrub-steppe are stable plant communities where diverse dicots can live with diverse monocots.