In the Japanese martial arts a distinction can be made between koryū 古流 (classical schools) and gendai budō 现代武道 (modern martial arts). Most modern martial arts come from old ryūha (schools / storm applications) that were practiced by the samurai in feudal Japan.
Lineage
The main difference between koryū and gendai budō is that in koryū the line of succession (from master to master) dates back to before 1868. This year was the start of the Meiji period, certain rights were abolished, for instance kirisute gomen 切舍御免 (samurai were allowed to kill lower-ranked people in case they had caused him loss of face) and certain laws were established. From 1873 laws came in effect no longer permitting samurai to wear swords in public.
Within koryū, a distinction is sometimes made between schools that were established before the Tokugawa shogunate (from 1600) and the schools were established within the Tokugawa period.
Jutsu and dō
Although there are many exceptions, the disciplines of koryū are often called jutsu 術 (technique) and modern schools do 道 (the “Way”). This is partly because the difference in intent of classical schools with respect to modern martial arts. Old schools were intended to prepare a practitioner for an actual fight. Modern martial arts practitioners have less need for their art to be used in actual combat, so there is room for personal development of the practitioner, through the martial techniques.
Weapon arts
Because samurai always carried weapons (especially the sword in the Tokugawa period) the weapon arts were more prominently represented. Even “jujutsu or taijutsu” arts of the time, often made use of short weapons or at least assumed that the opponent was carrying a weapon. Today there are many martial artist in modern schools who focus solely on unarmed arts.
Okuden
Unlike modern martial arts, old schools often have a secret part of the curriculum. Modern schools (in some cases) base their teachings on modern pedagogy, in which openness and sharing of knowledge is at odds with the idea of secret techniques.
Dan or Menkyō
The use of dan-degrees is a modern practice. Nowadays there are koryū that also make use of kyu/dan-degree system, sometimes in addition to an existing menkyo system, sometimes instead of a menkyō system. However, there are few modern martial arts (non that I know of) that have applied an traditional menkyō-system.
School or art
In modern martial arts, the emphasis is often on the (standardized) art. There is a form in which the martial art is exercised. There is only one kind of jūdō and kendō*. With traditional schools, there is no standardized form. Kenjutsu by itself is not saying much, without knowing in which school this martial art is practiced.
Background
Classical schools often have an origin in a particular province and period. Knowledge of the cultural, topographical and historical background of ones own school (and other schools) contributes to the understanding of the practice. Modern schools are easier to practiced separate from their origins. For example, a jūdōka do not need to know who the founder is to play well in matches.
