I was literally born for this question! I did my thesis on the 12th century Byzantine army!

Very different in some ways, but in others very similar.

I am fully aware that this answer is both accurate and completely unhelpful.

The Byzantine army in the 10th and early 11th century was based on a mix between a standing force based around the capital called the Tagmata. This consisted of both cavalry and infantry, and being a standing force was most likely well armoured and equipped. The other part of the army was provincially based. A Byzantine province at this time was called a Theme, and soldiers had their own land there which provided the income for them to obtain their own equipment and allow them to train. Such solders were essentially a reserve to be called up for campaigns as required.

By comparison, the Komnenid Byzantine army seems to have been a standing force, with troops paid directly by the central government and stationed both in the capital and in the provinces. This excerpt from the Alexiad of Anna Komnena supports this:

"Whilst on his way to Thessalonica, where the contingents from all the various districts were assembled, he thought it would not be amiss to hold a review of the troops drawn up in battle array."

The reference to troops from various districts seems to refer to garrison troops rather than thematic reserves as they were called up straight away on a campaign. There is another reference to troops being installed as a garrison rather than as a militia holding land:

"The Emperor had already thought of this and anticipated Bohemund's plan by sending the eunuch Eustathius (whom he promoted from the rank of Canicleius {= The keeper of the red ink used for imperial signature} to Great Drungaire of the fleet) with orders to occupy Curicum with all speed. Further he was to rebuild it quickly, and the fort Seleucia as well, which was six stades distant, then leave an adequate garrison in each and appoint Strategius Strabus Duke over them, a man of small body, but of long and varied military experience."

However, there was a trend in the late 11th century to move away from the thematic system and switch to paid, full-time garrisons, so the Komnenid army should be seen as the fulfilment of this process, rather than a straight departure from the thematic army.

There is a common assumption that the 12th century Byzantine army was reliant on foreign mercenaries. Complete bullocks. The bulk of the army was mostly natives, with West Europeans mainly operating as heavy cavalry in certain elite regiments, and some levied troops providing specific services like horse archers. The emperor John Komnenos settled the defeated Kumans within the empire on condition of them providing soldiers when required.

In terms of weapons and armour the only major changes was that Byzantine heavy cavalry was rearmed and retrained to fight in the West European tradition, as described by the Byzantine writer John Kinnamos.

Otherwise, in terms of organization, tactics and weapons the army was not so different from the 10th century version. It still used heavy-armed troops with lights troops in support, it maintained both light and heavy cavalry and still used fortified encampments when on the march. All the descriptions in the works of Anna Komneoa, Niketas Choniates and John Kinnamos of equipment and troops in action match those found in military manuals like On Skirmishing and Campaign Organization and Tactics.

Sources:

The Alexaid, by Anna Komnena:

http://www.yorku.ca/inpar/alexiad_dawes.pdf

The Byzantine Empire 1025-1204: A Political History, by Michael Angold

Deeds of John and Manuel Comnenus, by John Kinnamos

The Development of the Komnenian Army 1081-1180, by John W Birkenshaw

O City of Byzantium, by Niketas Choniates

Three Byzantine Military Treatises, translated by Dennis George