Question:

There seems to be a contradiction between the hadith of fasting on the 10^th of Muharram and the hadith that encourages free spending on the same day?

Answer:

There is no contradiction according to this humble one.

1. The hadith of spending freely on one’s family (dependants) may mean to purchase the items in the daytime and eat it after sunset. Mentioning the asbaab in place of the musabbibaat is a norm.
2. Another possibility is that this hadith is directed to the non-baaligh children who do not fast. The word “dependants” is an indication to this.
3. The statement of some scholars indicates that they take the hadith to mean spending on the night of Aashurah. The great Maliki Jurist: Abdul Malik ibn Habib (d. Ramadhan 238AH) said:

لا تنس لا ينسك الرَّحْمَن عاشورا              واذكره لا زلت فِي الأخيار مذكورا

قَالَ الرَّسُول صلاة اللَّه تشمله                     قولا وجدنا عليه الحق والنورا

منْ بات فِي ليل عاشوراء ذا سعة                         يكن بعيشته فِي الحول محبورا

فارغب فديتك فيما فِيهِ رغبنا                      خير الورى كلهم حيًا ومقبورا

              However all the narrations of this hadith used the word “يوم” (day).

        4.  “Spending freely” concerns general expenditure (نفقة). So on this day spend freely on                        those people whom one has to usually take care of throughout the year.

Therefore there is no contradiction between the hadith of spending freely on one’s dependants and the hadith of fasting on the day of Aashuraa.

And Allah Ta’aalaa knows best.
(Ameerul Mumeneen fil Hadith Muhaddithul Asr Mualana) Muhammad Yunus Jonpuri (May Allah preserve him)
Extracted from “Al-Yawaaqeetul Ghaaliyah” 1/206
Also see here for status of the hadith under discussion.