I think scientists have chosen not to define these terms more precisely because they are highly subjective and vary depending on the location and system being studied. Therefore, scientists for instance, prefer to use the terms in a broad sense, as it allows them to also consider localized and regional nuances when considering the interactions of organisms and their environment.
The shortest and least technical definition of 'ecology' is: 'Environmental Biology'.
The most appropriate towards the end of the 20th century is: 'The totality or pattern of relations between organisms and their environment'
And what with Biomes?
Regional climates interact with regional biota and substrates to produce large easily recognizable community units, called 'biomes'! A biome is the largest recognizable land community unit. It is the 'lifeform' of a uniform climax vegetation for a certain climate.
Read more in the 'Fundamentals of Ecology' by Eugene P. Odum (ISBN 0-7216-6941-7) (1971).
Thanks Professor Odum for defining and describing 'ecology' and 'biomes' half a century ago!
While both terms describe the interaction between the organisms and the environment, the size of the two may matter. The size of an ecosystem can be as small as a drop of water, or the whole earth can be considered an ecosystem. However, we have many biomes on earth, and there may still be multiple miniature ecosystems in each biome.