A summary is a brief account of something longer. Summery is an adjective meaning suggestive of summer — a summery dress. The A is the précis; the E is the season. Spellcheck accepts both, so the swap survives.
The core difference
A noun about brevity and an adjective about the season.
- summary — a brief account: an executive summary. Also an adjective: summary dismissal.
- summery — like summer: a summery afternoon.
The other summary
Summary as an adjective means done quickly and without formality — summary judgement, summary execution. That is a precise legal sense meaning without full proceedings, and it surprises people who know only the noun.
The spelling
Summery follows the usual pattern for making an adjective from a noun: wintry, watery, silvery. Summary comes from Latin summa, the sum — the whole reduced to its total.