As above varies a lot between subjects. For medicine Oxford is at least on the surface "harder" to get into in terms of success rates, but Cambridge has additional requirements which thins their field through self-selection at the outset. For chemistry Oxford is on the surface potentially easier to get into as it has a higher success rate than Cambridge natsci but the latter does not distinguish between the intended focuses of the applicants so it may be those planning to specialise in chemistry in natsci do just as well as those applying to Oxford.
Generally they're looking for essentially the same kinds of students. They just assess those somewhat differently in the admissions processes so you may present a better application to one than the other in different circumstances (or find an offer easier or harder to achieve after getting one). Broadly speaking Oxford put more emphasis on GCSEs while Cambridge puts less emphasis on these, while Oxford has lower standard entry criteria for A-level grades (or equivalent) while Cambridge are somewhat higher. They also may have different admissions tests (e.g. MAT vs STEP for maths), or a subject may require an admissions test at one of the two but not the other.
Also bear in mind each has a number of courses unique to them that is not offered by the other, either in general or in that particular format (e.g. arch & anth vs archaeology, ASNAC, land economy, vet med, human sciences, natural sciences vs individual science subjects, PPE, HSPS, etc, etc).