Sodium azide reacts with 1-bromobutane about $10^3$ to $10^5$ times faster in DMF than in methanol. The main reason is that

1. ADMF ionises 1-bromobutane to a carbocation, which then reacts with azide
2. BDMF has a much lower dielectric constant than methanol, so $\mathrm{NaN}{}_3$ is completely ion-paired and unreactive in methanol
3. Cmethanol hydrogen bonds to azide ion and lowers its energy, whereas DMF solvates $\mathrm{Na}{}^{+}$ but leaves azide poorly solvated and more reactive
4. DDMF hydrogen bonds to azide ion more strongly than methanol does, which makes the azide more reactive