Depreciation is the allocation of the cost of a tangible fixed asset over its useful life. Amortization is the allocation of the cost of an intangible asset over its useful life. In both cases, there is no cash flow associated with these assets after the year of purchase and so should not affect your cash flows in a DCF valuation.
However, both depreciation and amortization become relevant in a DCF valuation because the tax rules in many countries allow them to be treated as an expense reducing taxes. This tax deductibility impacts their cash flows as a result of the tax shields they provide.
If both depreciation and amortization are tax deductible there is no difference from a DCF valuation perspective. However, not all amortization is tax deductible and so not relevant from a DCF valuation perspective. Not only is not all amortization tax deductible, but the rules also vary between countries.
Therefore, the difference in depreciation and amortization kicks in when there is a difference in the tax treatment between the two in the countries’ tax codes.