These terms refer to totally different markets.
Carbon credits refer to the absorption or avoidance of greenhouse gas emissions in the voluntary carbon market. For example, by planting a tree (absorption) or switching from a wood oven to a photovoltaic oven (avoidance). Carbon credits are sold by the project owner to an individual or a company that wishes to offset its emissions. This is an unregulated market that has faced its share of controversy.
Emission allowances are standardized units of exchange on regulated carbon markets. They represent the right for an industrial polluter within the emitting market to emit one ton of greenhouse gases. Their supply is capped, equivalent to the market's carbon budget. This supply is progressively reduced, which gradually increases the value of each allowance and reduces emissions in a quantified and demonstrated manner. Emission allowances are regulated, liquid financial instruments, issued by jurisdictions, and traded on massive markets.