"Weaker" is not the right word for this; what you mean is "lower yield," which is a technical term that lacks a pejorative implication. Because as you can guess, neither the US nor USSR/Russia would adopt any technology they thought was "weaker"!

The basic answer is that nuclear warheads can be engineered to maximize different properties. You could, if you wanted to, optimize them for maximum yield. That certainly has a scare factor to it, and can do a lot of damage. But high yield weapons were also high-weight and high-volume weapons. They were hard to "deliver": to get them to their targets reliably and successfully. The famous Tsar Bomba is a classic example of that: a nuclear weapon of enormous explosive power (100 megatons at its full strength), but so large that it had to be slung underneath a heavy bomber like an egg. Such a weapon will be relatively slow to get to its target, and will be vulnerable to being shot down the entire time, but it is putting a lot of your stockpile "eggs" into one "basket." It's an expensive weapon that isn't really going to have a great chance of actually playing a part in the war.

Let's imagine, for example, that you had a target that you really wanted to drop a Tsar Bomba on. How important for you is that outcome? Let's say you want 90% certainty that a Tsar Bomba would go off on it. But let's imagine you think that every bomber with a Tsar Bomba attached to it has only a 50% chance of actually succeeding in their operation (which is a pretty optimistic random number I chose — remember that the US, NATO, etc., has invested in anti-bomber jets, anti-bomber radar systems, anti-bomber nuclear-tipped surface-to-air missiles, etc., along with the possibility of mechanical error with the plane or the bomb itself). Without getting into the math, to get a 90% certainty with something that has a 50% chance of succeeding requires sending at least 4 bombers against the target. If your chance of success drops to 25%, you need to send 9. If you drop your desire for certainty to something like 50%, and have a 50% chance of each one making it, you could get away with one... but if you don't care about it that much, why are you wasting a bomber on this? These are just example numbers, but you can see that this is a pretty inefficient way to do things.

Separate from this is another problem. The damage done by nuclear weapons does not have a linear relationship with their yield. That is, if bomb A has an output of 100 kilotons, and bomb B has an output of 1,000 kilotons, bomb B is not 10X more damaging than bomb A. The distance of the blast and thermal effects of an explosion scales as a cubic root. So a given blast effect from bomb B is only going to go a more than 2X as far as the same effect as bomb A. That doesn't mean that bigger bombs aren't indeed more destructive — but that you see something closer to a doubling of range of effects as you increase the order of magnitude of the bombs. (You can experiment with NUKEMAP to see what this means intuitively.)

But weapons of comparable design sophistication do have their weight scale linearly as a function of their yield. So that 1,000 kt bomb will weight 10X more than your 100 kiloton one. So as you increase the yield, you get diminishing returns in the form of damage, but increasing penalties for weight and volume. Hence the ridiculous unwieldiness of the Tsar Bomba. There are, to be sure, better and worse ways to design a nuclear weapon, and the Tsar Bomba is definitely heavier than it needed to be for that yield. But the general rule stands.

So what happened? In the early 1950s, the US and USSR both raced to make very high-yield bombs. This was driven both by their "horror factor" but also by the fact that their ability to deliver the weapons accuracy was pretty limited. You can compensate for poor accuracy by having a bigger nuke (it makes it hard to "miss"). By the early 1960s, though, they figured out a) how to make relatively compact weapons that were in the roughly 100-1000 kiloton yield range, and b) how to make more accurate delivery systems. They also realized, by the late 1960s-1970s, that the real "name of the game" for nukes was to have things like MIRVed missiles (long-range missiles that could each carry multiple warheads on them, up to a dozen or more per missile) that could overwhelm missile defense systems, or cruise missiles (that could evade systems). To do that well you want a premium on compactness of warhead, and the "sweet spot" for warheads of the right size and weight turns out to be around 100-500 kilotons. So they worked to make those systems super accurate as well, and more and more reliable, so that they could have an arsenal of a fixed size but would be guaranteed the ability to create an unacceptable level of destruction.

The higher accuracy meant that they could destroy any target they wanted to with a high reliability; even if you did want to destroy an entire metro area, it is easier to do that with 5 warheads loaded onto ICBMs or submarines than it is to do it with one big, slow, clunky, heavy warhead. And whereas your big, slow, clunky, heavy warhead really has only one imaginable use (destroying a metro area, or excavating an underground bunker complex), your smaller nukes could be reconfigured to different targeting scenarios as you saw fit (so you could use them to target other nuclear weapons silos, for example, without "wasting" them).

Anyway, all of which is to say, the weapons that got developed in the late Cold War, and are still being used today, aren't "weaker." They are actually far more deadly by several measures, even if they are less powerful. They are more reliable, they are more flexible, they are more accurate, they are more efficient, they are more credible as weapons you could expect an enemy to think you might be able to successfully use, etc., than the old Cold War monster bombs that could barely (or not even barely) fit inside their bombers.

For further reading: Eric Schlosser's Command and Control is a great general-audience history of nuclear weapons development over the Cold War that is very readable, focused around issues of safety and accidents, but generally applicable to this kind of question. Donald MacKenzie's Inventing Accuracy is a great sociological history of nuclear weapons accuracy issues that gets into the technical and political dimensions of exactly the kind of late Cold War (1960s onward) shift I am talking about here. And I wrote an article a year ago about the development of the Tsar Bomba, and the response by the US (which included looking into developing its own weapons of comparable yield, but abandoning that search), which touches on many of these same issues.