The theoretical underpinnings are very similar. The Explanatory Tantra (bshad rgyud) that describes embryology, physiology, conditions of disease etc, is largely identical to the āyurvedic text the Aṣṭāṇgahṛdayasaṃhitā (AHS), appearing only after the translation of the AHS into Tibetan by Rinchen Zangpo. The largest differences are in the diagnostic methods, religious bases, and in their respective pharmacopeias.
Pulse diagnosis in Tibetan Medicine appears to predate its Āyurvedic counterpart, and makes use of the 5 phases found in Chinese Medicine (Fire, Earth, Metal, Water, and Wood) to draw connections between the internal organs and external seasons. There are also differences in urine analysis.
As far as the religious underpinnings go, both use the common Indian 5 elements (pañcabhūtā) as the building blocks for the three doṣa-s, the tastes etc. Tibetan Medicine does not make use of the tattvas of the sāṃkhya system, instead using Dharma as a foundation. Āvidyā gives rise to the three poisons of desire (rāga), hatred (dveṣa) and delusion (moha), which in turn give rise to the three doṣas (vāta, pitta, kapha).
Being located inbetween China and India, Tibetan Medicine makes use of a range of medicinal substances, indigenous, Chinese and Indian.
There are other differences in how some of the treatments are now done, but the differences above seem to be the most prominent.
Wishing you all the best.
སེམས་རྣམ་པར་གྲོལ་བར་བྱའི་ཕྱིར་བྱམས་པ་བསྒོམ་པར་བྱའོ།“In order to completely liberate the mind, cultivate loving kindness.” -- Maitribhāvana Sūtra