Advance and progress both as intransitive verbs and as nouns share the meaning to move (or movement) forward in space, in time, or in approach to a material or ideal objective. They are often employed interchangeably; however there are instances in which one is preferable to the other.
Advance only may be used when a concrete instance is signified; though one may say that at a given time science made no advance (or progress), one must say that there were no advances (not progresses) in science at that time.
Advance is preferable to progress when the context implies movement ahead such as that of an army marching to its objective, the distance traveled, or the rate of traveling,
- bullish sentiment regained fervor . . . and stock prices advanced sharply
—N. Y. Times - there are some . . . who picture to themselves religion as retreating . . . before the victorious advance of science
—Inge - boll weevil . . . may have existed in Mexico . . . for centuries . . . it advanced north and east at the rate of about 100 miles per year
—Harlow
Progress usually carries implications derived from earlier meanings of a process, a circuit, or a cycle, and so is preferable to advance when the movement forward involves these implications, as by suggesting a normal course, growth, or development.
- the trial is progressing
- moon . . . begins . . . her rosy progress
—Milton - [summer] oft, delighted, stops to trace the progress of the spiky blade
—Burns
Sometimes the word without losing these implications carries additional connotations and often stresses development through a series of steps or stages, each marking a definite change.
- it would be . . . a dull world that developed without break of continuity; it would surely be a mad world that progressed by leaps alone
—Lowes - the progress of an artist is a continual self-sacrifice, a continual extinction of personality
—S. Eliot
Progress is the preferable word when development with improvement is implied.
- there was a general belief in inevitable and universal progress
—Berger