Both dynasties experienced major catasthrophies.

Wang Mang usurped the throne, marking the end of the Western Han, reigned for 14 years, and was toppled. The Han dynasty was revived and lasted for almost 200 years after that, known as Eastern Han dynasty.

The Tang dynasty also got interrupted briefly by Empress Wu ZeTian who was the sole Zhou monarch for 15 years before power was return to her sons who restored the Tang Dynasty. Compared to the rebellion of An LuShan half a century later, it was a minor hiccup.

The real crisis of the Tang was the An LuShan's rebellion in AD 756 broke the power of the Tang dynasty. Though the dynasty lasted for another 150 years until AD 907, the dynasty did not have the kind of post-crisis recovery experienced by the Eastern Han dynasty.

Why the different fates?

In comparison to the Han when no Liu clan was emperor during the rebellion of Wang Mang, the Tang emperors continued to rule despite being forced from the traditional capitals of ChangAn and LuoYang. Much of the country remained loyal whereas the Liu clan of the Han dynasty had to start practically from scratch.

My personal assessment is that the answer could be found in the different social structures and makeup between the Han dynasty and Tang dynasty.

During the Han dynasty, power was distributed to landed hereditary aristocrats and gentries whose support was essential. With their background, these factions tend to be big on traditions, conservatives, and orthodoxy. The mainstay of their power were the farmlands, not urban cities.

In contrast, the power during the Tang dynasty was distributed to military governors who ruled from cities. Many came from humbler background as nobilities in the Tang had been shunning military service. The military governors had little respect for orthodoxy and traditions, many were unlearned and came to their positions through a career of violence.

Another factor was the presence of numerous non ethnic-Hans in the Tang, and the Tang's reliance on them for the military. When these tribes found the Tang no longer worthy of respect, they turned their skills against the empire for their own benefits.

I also considered significant that the primary restorer of the Han dynasty, Liu Xiu, rose through the ranks and personally led in combat before finally emerging victorious and establishing himself.

In contrast, Li Heng, who succeeded Emperor XuanZong, was known more for his poetry and had relied on loyal generals to fight his battles. Thus, his influence over the military was limited. Worse, he died before the rebellion was actually put down and the work of consolidation was left to his successor Li Yü (李豫) whose posthumous name was DaiZong. These rulers lacked personal prestige compared to Liu Xiu who was able to implement some reforms and strengthen his dynasty.

I would like to hear your opinions on these.

I think another major distinction is the caliber or background of the ursurper. Wang Mang's reforms were actually quite good if allowed to be carried to fruitation. He was already pretty much holding the reins of government even before he usurped the throne. The Han dynasty was already in serious trouble then and Wang Mang was seen by some to bring back order to the empire. An Lushan on the other had was a war general. He was influential no doubt but never the less, he was seen more as a warlord and not a serious alternative to the ruling family.

The situation in mid-Tang was more similar to that at the end of the Eastern Han, after the Yellow Turban Rebellion. In both cases, the central army had been too weak and regional military commanders and governors had had to raise armies on their own. After peace was restored, they refused to hand over their power and armies, and instead began setting themselves up as warlords, passing on their positions to their sons.

Liu Xiu was able to avoid such a situation because his rivals were all small-scale rebel leaders like himself, rather than generals. He was able to defeat his most formidable rivals the Red Eyebrows, and he was a more capable leader than any of the late Han or mid-Tang emperors.

I don't think it applies to his currency policies.

But how would you relate it to the the Han managing to reinvigorate itself while the Tang continued in its extended death throes?

I can't really see social order was less disrupted during Wang Mang's reign compared to An LuShan's rebellion - according to wikipedia, between AD 2 and AD 5 and again in AD 11, the Yellow River changed course to flow south (instead of north) of the Shangdong Peninsula, causing famine, epidemics, and migration among the peasants.

But Liu Xiu himself was also not a major power. In fact, he was relatively minor and nearly got killed by his master, Liu Xuan. Nonetheless, by personal participation in battles, he won over more lasting submissions than the Tang emperors who stayed mostly within their palaces.

There were certainly a lot of similarities between mid-Tang and post Yellow Turban rebellion/Dong Zhuo's Eastern Han as you mentioned.

But I'd also look at Liu Bei as someone who could have replayed Liu Xiu's accomplishments had he been more fortunate.

Because of this, I could not see how the regional separatist regimes during post-An LuShan Tang dynasty were more powerful or entrenched than the rivals faced by Liu Xiu or Liu Bei.

I could only put it down to the misguided policies of the later Tang emperors not to more aggressively campaign for control, as Liu Xiu and Liu Bei did. They did not dare to stake whatever's left of their dynasty.

Probably also because the external threats occupied much of the dynasty's attention - Tibetans and Uyghurs were constantly giving them trouble until the 840s.

Tang Wuzong and Tang Xianzong did attempt to restore central control through successful military campaigns. But the shaky loyalty of the military governors put all their efforts to waste. As soon as they were not given enough rewards, or were threatened with a loss of autonomy, they would rebel against the government even if they had previously fought on its side against other rebel governors. And while they would fight each other much of the time, they would also form coalitions against the central government if they perceived that the emperor was aiming to threaten all of them. It was a balance-of-power game that favoured the opportunists rather than the predictable imperial court.

What about the roles of eunuchs?
Both dynasties had eunuchs becoming influential, but it seemed that the Tang eunuchs were more powerful than the emperors. How did that happen?

Was there a viable policy that the Li imperial clan could have adopted to regain their authority?

Tang eunuchs first became powerful during the suppression of the An-Shi Rebellion, with the likes of Li Fuguo and Yu Chao'en being the favourites of Suzong and Daizong respectively, and eventually becoming so domineering that the emperor had to have them killed. Before that, their power was relatively benign, for example Tang Xuanzong's head eunuch Gao Lishi.

But in Gao Lishi's time, eunuchs could already command units of the imperial guards. Eventually, the Army of Divine Strategy (Shence Jun) was always commanded by a enunch. And since the officials in the Tang court later were split by factional fighting, the eunuchs became the only regular fixture of the court and the ones who had the most influence on the emperor. They murdered or deposed quite a few emperors, including Xianzong.

Wang Mang is more comparable with Empressess Wu's reign than with An Shi.

The Tang was basically slowly reduced to the status of the Eastern Zhou after An Shi, it's real central power decline ever more rapidly.