The main difference here has to do with the time frame that each method of analysis looks at. A horizontal analysis typically looks at a number of years. By contrast, a vertical analysis looks only at one year.
A horizontal analysis compares financial information for one company with the same types of financial income for the same company in one or more previous years. For example, you could look at the company's inventory and determine the percent change for its inventory over each of the last three years.
By contrast, a vertical analysis is more of a snapshot. It shows all of the firm's financial information for a particular year. Each item on the statement is typically expressed a percentage of some particular statistic. In other words, you might express everything as a percentage of the firm's total assets. This form of analysis allows a firm to compare itself quite easily to other firms in its industry.