Lets start with the similarities. Worship the gods and goddess of the Aesir and Vanir, perform rituals where some sort of material offering is given to the gods (mead, grain, incense etc), honors ancestors and heros of the folk, honors deed done by modern heathens that are of note, study "lore" to gain a better knowledge of our ancestors and how to properly worship the gods and quite a few more.
There are charicteristics of note in Theodism which are typically NOT true of Asatru: sacral lordship, hierarchal tribal structure, thew (custom) as the basis of ritual practice ethics etc, the lengthy period of labor and learning prior to being accepted as a full member of a theod- normally referred to as "thralldom" and the practice of animal sacrifice.
The main difference is in how theodsmen approach Germanic heathen religiosity as a community rather than a group of individuals that all worship the gods. In Asatru you can have an individual Asatruar that is not part of any group and worships entirely on their own. There is no such thing as a solitary Theodsman. If you aren't part of a theod (tribe) then you aren't a theodsman (tribesman). One may follow Theodish thew (custom) but is not a theodsman.
Theodism is at its core a communal religion. When we do a rite it is a form of communication with the gods. The folk in a theod bind ourselves to one another in various ways; by oath, giving of gifts etc. We gather together in order to do a faining or blot; we are communicating various things to the gods like love, respect, giving of sacrifices etc. In Theodism, the theod as a whole sends a single "message" to the gods in the form of a ritual. The gods in turn impart their gifts to us as a whole unit. In Asatru, ritual is designed to allow each individual send their own personal "message" to the gods. So the difference becomes most clear in ritual practice. In an Asatru "blot" there is lets say 20 people who each say something over the horn and thus sending twenty different messages to the gods. Each individual is of equal status and their communication to the gods seen as being just as good as any other and everyone in the group having a turn.
In Theodish ritual, there is one person of the highest status within the Theod (usually, the lord of the Theod or a priest), representing the whole, performing the ritual, usually in an old language (Anglo-Saxon for example) and giving sacrifices on behalf of the entire theod (tribe), thus sending one message.
Despite Swain's revisionism, the Ealdriht was NOT a Theodish group. Swain only claimed it to be so very late in it the existence of the Ealdriht and after it broke up. Unfortunately, since Swain is the squeaky wheel of Theodism he gets the most attention. No one who is a Theodsman outside of Swain, thinks of the Ealdriht as having been "theodish." Both "theods" that came from the defunct Ealdriht have imploded.
Theodism began as an Anglo-Saxon tribalism but is now represented by a number of different tribal thews. Frisian, Norman, and Jutish for example.
The Way of the Heathen by Garman Lord Gering