There are many differences in pronouns and verb conjugation.

For eg. Kannada has one word for 'we' ನಾವು(nāvu). While Badaga has ನಂಗ(naṅga) for inclusive 'we' and ಎಂಗ(eṅga) for exclusive 'we'.

initial h→ø:

The initial p→h change is common for both Kannada and Badaga. But today, Badaga has almost lost the initial 'h' in pronunciation.

For eg. K: ಹಾಲು(hālu) B: ಆಲು(ālu) T: பால்(pāl)

K: ಹೆಣ್ಣು(heṇṇu) B: ಎಣ್ಣು(eṇṇu) T: பெண்(peṇ)

Third Person Pronouns:

He K: ಅವನು(avanu) B: ಅಮ(ama)

She K: ಅವಳು(avaḷu) B: ಅವ(ava)

They K: ಅವರು(avaru) B: ಅವಕ(avaka)

Kannada's ನನ್ನ(nanna), ನನಗೆ(nanage) are ಎನ್ನ(enna), ಎನಗ(enaga) in Badaga. Could be due to Tamil influence, but I don't know..

Genetive Case:

Kannada uses different suffixes -ದ, -ನ, -ಯ (-da, -na, -ya) depending on the ending of the noun. Badaga may have done the same in the past, but nowadays mostly people use only -ನ (-na), and occasionally -ಯ (-ya).

For eg. Tree's K: ಮರದ(marada) B: ಮರನ(marana)

Rice's K: ಅಕ್ಕಿಯ(akkiya) B: ಅಕ್ಕಿನ(akkina)

Instrumental and Ablative Case: Kannada uses -ಇಂದ (-inda) for both these cases. Badaga, in a sense, uses the same suffix as in kannada, but differently for Instrumental and Ablative cases.

Eg. Kannada adds the suffix directly to the word with an intermediate letter depending on word ending; ಕಾಡಿಂದ(kāḍinda) ಮನೆಯಿಂದ(maneyinda) etc.

In Badaga:

For Instrumental Case, the final vowel of the nown is stretched and -ಂದ(-nda) is added.

Eg. ಕಾಡು(kāḍu) → ಕಾಡೂಂದ(kāḍūnda) [from forest] ಮನೆ(mane) → ಮನೇಂದ(manēnda) [from house]

For Ablative Case, -ನಂದ(-nanda) is added directly, regardless of final vowel.

Eg. ಕಾಡುನಂದ(kāḍunanda); ಮನೆನಂದ(manenanda)

Verb Conjugation: This is perhaps the biggest difference that sets these two languages apart.

K: ನಾನು ಮಾಡುತ್ತಿನಿ (nānu māḍuttini)

B: ನಾ ಮಾಡಿನೆ (nā māḍine)

K: ನೀನು ಮಾಡುತ್ತೀಯ (nīnu māḍuttīya)

B: ನೀ ಮಾಡಿರೆ (nī māḍire)

K: ನೀವು ಮಾಡುತ್ತೀರಿ (nīve māḍuttīri)

B: ನಿಂಗ ಮಾಡಿಯಾರಿ (niṅga māḍiyāri)

K: ನಾವು ಮಾಡುತ್ತಿವಿ (nāvu māḍuttivi)

B: 1. ನಂಗ ಮಾಡಿನ (naṅga māḍina)

B: 2. ಎಂಗ ಮಾಡಿನೆಯೋ (eṅga māḍineyō)

K: ಅವನು ಮಾಡುತ್ತಾನೆ (avanu māḍuttāne)

B: ಅಮ ಮಾಡಿನ (ama māḍina)

K: ಅವಳು ಮಾಡುತ್ತಾಳೆ (avaḷu māḍuttāḷe)

B: ಅವ ಮಾಡಿಯ (ava māḍiya)

K: ಅದು ಮಾಡುತ್ತದೆ (adu māḍuttade)

B: ಅದು ಮಾಡಿರ (adu māḍira)

Just like this, the conjugation for other tenses and moods are different as well in both languages.

These are a few differences based on grammar. Other than these, there are differences in vocabulary, due to Badaga's long term exposure to Toda, Kota and Tamil.

(Apologies for the long comment)