I don't know enough about agriculture specifically to talk about the daily work life of a farmer in either era, but I imagine there would've been stark differences in population density, more widely shared cultural mores, and generally more developed infrastructure connecting the various kingdoms.

77BCE predates the Roman conquest so a farmer living in this era would have had a much more limited understanding of the broader world. You said 'apart from religion,' but in many ways it's difficult to separate religion from the daily life of a person living at the end of the first millennium, at least insofar as how joe farmer would imagine his place in the world. By this time the mark Rome left on the island would've been unavoidable (speaking of physical remains), and when the Normans conquered the island just a few years on, the influence of that reign and the The Roman Catholic Church would leave marks that still can be seen today.

The tribes that inhabited the islands in 77 BCE certainly had dealings with the Mediterranean world, to say nothing of the inter-tribal dealings and shared cultural heritage, but by 1059 the intervening centuries saw so many changes through both the precipitate social upheaval brought on through war, and also more gradual influence through trade and the concomitant spread of ideas that I would imagine a tribal farmer in pre-roman Britain would have a more local understanding of his environment. --- I don't mean to imply that a farmer in 1050 would care in the slightest about anything beyond his local existence, rather that he would be more likely to have heard of Jerusalem or Constantinople, say. And that, theology aside, even a passing familiarity with Christianity carried with it such dense cultural weight that it would be near impossible to have it not color your experience and perspective in some way.

I think that the English Saxon farmers would also be involved in the wool trade, while the celtic farmer most likely would not. I believe wool had become an important industry by the middle ages, and that many Saxon farmers depended on their wives to convert wool into textiles (for trade with France) as apart of their livelihoods.

Source: Robert Bucholz, History of England from the Tudors to the Stuarts. Great Courses No. 8470.

One of the biggest changes in agricultural technology would have been the introduction of the mouldboard plough.

The Celtic farmer in 77 BCE would have used a "scratch plough" (or "true ard") - Quite possibly pulled by oxen.

The Saxon farmer in 1059 AD would have used a mouldboard plough almost certainly pulled by oxen.

The mouldboard plough had two advantages. It greatly reduced the amount of time to prepare a field, allowing a farmer to farm more land and grow more produce. In the form of the heavy wheeled mouldboard plough (sometimes known as the "Roman Plough"), it also allowed tougher and more difficult soils to be ploughed and put into agriculture.

There is archaeological evidence that the Roman plough was used in Britain by about the late 3rd or early 4th century AD.

By the 8th or 9th century, the three field system (one field planted in grain, one in other crops (usually peas, lentils or beans), and one left fallow as pasture)- rotate annually - had replaced the old two field system (half the land left fallow every year) in Britain. (The secret to it's success was planting the third field in nitrogen fixing crops, which helped renew the soil.)

These two changes meant that the Saxon farmer could put more land in production and produce more crops (and larger surpluses) than his Celtic predecessor.

The replacement of oxen by horses as the main draft animals for ploughing came later in Britain (12th century?) and this led to another improvement in farm productivity, as horse teams can plough faster than ox teams. (Raising Crops in Northern Europe, with a short growing season, means that there is a limited time between the ground thawing and the crops needing to be planted, in which ploughing can be done. What cannot be ploughed in that window cannot be successfully cropped that year).