First of all, the biggest difference is that emulsified asphalt has water and modified asphalt has no water.
Emulsified asphalt is a kind of emulsion, in other words, very small asphalt droplets are suspended in water. The continuous phase is water and the dispersed phase is asphalt. Asphalt exhibits solid like properties at room temperature and liquid like properties at high temperature. Therefore, the asphalt can be used at room temperature after being dispersed into water, which is the biggest advantage of emulsified asphalt. The manufacturing process of emulsified asphalt is shown in the figure below. In a word, it is to mix the asphalt heated into liquid with emulsifier, stabilizer, acid or alkali with colloidal mill to obtain emulsified asphalt.

The density of asphalt is very close to that of water, so asphalt can be suspended in water. Emulsifier is a kind of surfactant, which is a series with soap and washing powder.  One end of its molecule is hydrophilic and the other end is lipophilic. Once mixed with asphalt and water, it will be adsorbed to the interface between asphalt and water to wrap the asphalt. The hydrophilic end is charged, so that it can remain stable after emulsification. Emulsified asphalt is essentially the same as milk. Milk is a small fat particle wrapped by protein and suspended in water.
Emulsified asphalt looks like a dark brown liquid, as shown in the figure:

Modified asphalt is relatively simple. Pure asphalt is called base asphalt, so adding modifier to base asphalt will become modified asphalt. There are too many kinds of modified asphalt. The most common is polymer modified asphalt, such as SBS modified asphalt. Adding polymer can greatly change the properties of asphalt. For example, SBS can increase the ductility of matrix asphalt to more than 1.5m. The intuitive feeling is that asphalt can be pulled into wire.