Marriott Corp. yesterday reported only a slight increase in 1980 profits, although sales at the hotel, food service and amusement company rose by 14 percent.

Marriott reported 1980 profits of $72.03 million ($2.60 a share) compared with profits of $71 million ($1.95) for 1979. Marriott's sales rose to $1.71 billion in 1980 compared with 1979 sales of $1.5 billion.

For the 1980 fourth quarter, however, profits dropped to $16.9 million (64 cents) compared with 1979 fourth-quarter earnings of $17.3 million (51 cents), despite a sales increase to $542.6 million from $457.6 million for the same period in 1979.

Marriott President J.W. Marriott, prediciting that the company's sales would exceed $2 billion, said Marriott "fared well in difficult environment which affected some operations more than others in 1980."

For example, hotel profits rose by 17 percent as sales in that division rose by 21 percent. Restaurant profits were only slightly ahead of 1979 profits, while results at Marriott's theme parks were off slightly with a 5 percent drop in profits.

In other reports, Richmond, Fredericksburg and Potomac Railroad Co. reported 1980 profits of $15.7 million ($43.84 a share), up from 1979 profits of $13.7 million ($38.39).

Revenues rose to $49.6 million in 1980 compared with revenues for the previous year of $43.9 million.

For the fourth quarter, the company reported a slight increase in profits, as sales rose 9 percent to $12.8 million.