First off, the reasons that Belize became independent (as a possession of GB) are not causally related to why Spain's possessions became independent, so these are two very different explanations.

From what I've found (not an expert on British colonial possessions) Belize only became fully independent in the 20th century, being a colonial possessions with limited autonomy all through the 19th.

Regarding the fragmentation of Central America, the short answer is: lack of a united proto-national identity, plus lots of jealous local interests.

Central America was the Capitancy of Guatemala under the Spanish regime, being administratively dependent on the Viceroyalty of New Spain (ie. Mexico).

Central America was also different in that the dominated peoples, largely Maya, were different culturally and ethnically from the Aztecs and other indigenous peoples of Mexico. The elites, who are the ones who mattered anyway, were also endogenous and jealous of local privileges. EDIT: To be clear, the white or almost white elites. I mentioned the different ethnicities of most people in the region only to explain why the common people didn't identify too readily with Mexico, which was dominated by whites anyway.

When Mexico became independent in the early 1820s, as an attempt to preserve a conservative order, it was weakened by a weak central government, anemic economy, untrustworthy military, and the scars of several huge popular revolts. Central American elites saw this as their chance to break away from Spain as well, leading them to declare their independence from Spain as well. They initially pledged allegiance to the newly founded 'Mexican Empire' but a few years later declared themselves an independent republic, with the exception of a small territory which stayed loyal to Mexico.

Mexico sent troops to impose its rule in Central America, but the same year as Guatemalan independence, Iturbide (the Mexican Emperor) was dethroned and a coup placed a republic in power. The country was too unstable and too fearful of bigger priorities, like attempts at Spanish reconquest of Mexico, to focus on Central America, which was also distant from the Mexican Republic's its power base.

The following decades in the 19th century are a series of internal conflicts, crises, civil wars and separatist conflicts as every local elite successively seceded from the central government in Guatemala. Attempts at reunification were made, but failed. No national identity held them together, central government wasn't strong enough to impose order, and foreign powers did nothing to stop a fragmentation which made negotiations with local elites all the more easy. Nicaragua, for example, was a candidate for a canal for a while, so it was convenient that local elites would break off with Guatemala and thus weaken themselves when it came to bargaining with European powers.

Panama exists because the US aided an independence movement there with the express purpose of building an American controlled canal. We sent our navy there and made sure that the central government of Colombia would be unable to send aid to government loyalists fighting against Panamanian separatism. Theodore Roosevelt himself expressed it as 'I took Panama'.

Edit: Why are they still separate today? Nationalism slowly crept in and solidified while the area was fractured, and many of these countries have very different political views (some more liberal, others conservative, others semi-socialist) so it seems unlikely they will reconcile anytime soon.

I'd recommend reading this awesome book by Walter LaFeber which outlines the independence and evolution of each country up to the middle to late 20th century.

http://www.amazon.com/Inevitable-Revolutions-Central-America-Edition/dp/0393309649