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Both the US and the UK formerly had an upper legislative house which wielded considerable power despite being elected by government officials rather than the people (the Senate) or completely unelected (the House of Lords). In the early 1900s, both countries took actions to rectify this problem, but in radically different ways; the Senate was made directly elected, while the House of Lords was stripped of essentially all of its powers.

Why the great difference?

The two contexts are very different. The US is federal with the Senators representing the states at the federal level, while the UK was an entirely centralized state running an empire from Westminster.

The UK conflict over the budget which led to the Parliament act was effectively class warfare (with Winston Churchill on the progressive side!). Changing the structure altogether would probably have been out of the question. The US has a formal constitutional reform process, the UK does not; the compromise was only achieved by threatening to get the King to stuff the Lords with enough new Peers to vote it through.

(To me the part of the US system which matches the Lords most closely is actually the Supreme Court; significant power to make social change, but life appointments, and the only way to achieve democratic control is to appoint more people from a faction.)

Perhaps a more important question is why, given how often it's been in the Labour manifesto, the abolition of the Lords hasn't yet happened.

Because the UK simply couldn't have followed the US approach.

The US Senate was still quite young and democracy the official form of government, so these reforms were made easily.

In the UK however, the House of Lords was old, entrenched in tradition and - given that the official head of state is still the queen - removing the lords from the government would have brought a huge uproar among the influential (many of which held titles and were therefore on the "losing side" of this reform), so they did the most they could reasonably achieve: strip all the power while on paper retaining their title and (some) function

In the United States, the Senate exists to give small states more influence and counteract the greater population of the larger states. It is constitutionally enshrined in that role. At the time that the Senate was made directly elected, there was little support for reducing that role, particularly among small states. And remember, constitutional amendments require two thirds of the Senators to agree (or a constitutional convention, but that has never happened since the founding).

In the United Kingdom, the House of Lords was left over from a time when the lords had more power. The general trend has been to reduce that power. When support for reduction reached a sufficient level, it was done. Power switched from the House of Lords to the Commons.

In both cases, the change from unelected to elected would seem to be something of a side effect. First, the Senate was appointed by the democratically elected state legislatures. There were accusations of corruption in that process. Direct elections were considered less corrupt rather than more democratic. Second, the Lords were the legacy branch and the Commons the newer branch. Transferring power from the legacy to the newer branch is natural. It's merely an extension of the initial process.

The House of Lords did not exist because the Commons needed balanced. It would be more accurate to say that the Commons came into existence to balance the power of the lords.

The US solution, making Senators directly elected, would have made little sense in the UK. Rather than a natural change, it would have essentially eliminated the House of Lords and added a new House. But how would such a House have worked? There doesn't seem to be a movement in the UK to balance large population areas with small population areas. So what they would have had would have been essentially a second House of Commons, which would have been redundant.

TL;DR: while it is accurate to say that both switched from unelected to elected, the reasons behind the switches were quite different.

The US & the UK are two different polities with different histories. It’s not particularly surprising that they have different constitutional arrangements.

The key principle in the UK is the sovereignty of parliament and this is taken to mean that government is driven by the House of Commons where the legislature is directly elected. The House of Lords then effects oversight.

Its the same rationale that means that the US congress is bicameral, that is a legislative body that has two houses.