The functional differences follow from that. Donor systems handle recurring gifts as a first class concept rather than as subscriptions. They track soft credits, where somebody influenced a gift they did not personally make. They handle pledges, which are commitments to give later. They generate the acknowledgements and annual summaries that donors need for tax purposes, with the specific language required. They record restricted funds, meaning money given for a stated purpose that cannot be spent on anything else. And they track events, volunteering, and giving as connected activity by one person rather than as unrelated records.
That last point is the practical one. In a donor system, the person who attended in March, volunteered in June, and gave in November is visibly one relationship. In a general CRM those are three records in three places, and the pattern that would tell you who to approach is invisible.
Whether you need one in the first year depends on volume and complexity rather than principle. Below a few hundred donors with straightforward giving, a well structured spreadsheet plus whatever your payment processor provides is genuinely adequate, and several processors aimed at nonprofits include basic donor records. What pushes you over is recurring gifts at any volume, restricted funds, grant reporting, or a board asking questions your spreadsheet cannot answer.
If you are mission driven but not a registered nonprofit, and nobody is giving you tax deductible donations, you likely want a standard CRM. The specialised features exist to serve obligations you do not have.
Whatever you use, capture the source of every gift from the beginning, because attribution in fundraising is even harder to reconstruct than in commerce. Which appeal, which event, which person introduced them. Six months of that record tells you where support actually comes from, which is frequently not where the effort went, and it is the single most useful thing a small organisation can know when deciding what to repeat.
Check the export position before committing, since donor records are among the most sensitive data any organisation holds and the most consequential to lose access to. You want to be able to extract the full history including gift records and contact details in a standard format without assistance. An organisation that cannot move its donor list is dependent on a vendor in a way that affects its independence rather than merely its convenience.