While they had more rights than in other time periods and regions, women were certainly not 'legally equal' to men in the Old Assyrian period (men definitely were the legal and economic heads of the household, for example), but you are correct that there are some differences between the Old Assyrian (19th century BCE) and Middle Assyrian (12th century BCE) legal systems.

One of the main reasons for this difference may simply be that our sources are very different for those two time periods. The Middle Assyrian Laws are the first law code from Assyria that is actually preserved, so everything we know about the legal system in the earlier periods is derived from private letters and contracts, and not from any official proclamations (although we know for a fact that they existed already in the Old Assyrian period).

The private letters and contracts from the Old Assyrian period that are relevant in this context mostly concern marriage, divorce, and inheritance matters, as well as the role of women in the family business. As these documents were found in a trading colony in modern Turkey, far from the capital Assur, it is unclear to what extent they reflect legal practices in the capital – the economic situation of a woman whose husband is away on trading missions for most of the year or even residing in Turkey and her thus having to control the family's finances, for example, is very different to the situation of women from a different class.

The Middle Assyrian Laws are a law collection of conditional sentences giving crime and punishment (if ... then ...). Tablet A, which deals with women and family matters, includes topics such as adultery, rape, but also slander and false accusations. These are topics that only very rarely occur in the private letters and contracts from the earlier periods – probably because they are often criminal offenses and not private legal transactions.

The Middle Assyrian Laws very much frame women as legally dependent on their husbands or fathers, but we also see from the actual contracts (unfortunately, only a few are preserved for this time period) that in practice, women could partake in legal transactions on their own, even if they were married.

The one area of the law where we have good sources for both time periods is family law (marriage, divorce, and inheritance). Both parties are allowed to file for divorce in the Old Assyrian sources (and in many cases, they seem to file jointly) and daughters inherit part of their fathers' assets (although the evidence for this mostly comes from testamentary provisions, and the reason for the existence of these testaments might be that they are not in line with standard legal practice).

The Middle Assyrian Laws allow only for the husband to file for divorce and he does not have to pay compensation, but the two preserved marriage contracts from this period do allow for divorce by either party, so it is not certain to what extent these laws were really universal and applied in practice. There are also several Middle Assyrian contracts where we can see daughters inherit, although it seems that this was not the norm.

So, the bottom line is that there are some differences in the legal status of women, in particular with regard to family law, but they seem to be less pronounced in practice than it would seem from the law collection. Furthermore, many of the observed differences may be due to the differences in sources for the two time periods (private documents from far away from the heartland vs. a law collection as an official document).

And, to top it off, there is a period of 800 years between the Old and Middle Assyrian periods, and one that involved massive societal changes and population movements throughout West Asia, and this may very well also have affected the legal system.

Some literature:

For women in the Old Assyrian period: C. Michel: Women of Assur and Kanesh. Texts from the Archives of Assyrian Merchants. Writings from the Ancient World 42. Atlanta, GA: SBL Press 2020.

For the Middle Assyrian Laws (and other law collections): M. Roth: Law Collections of Mesopotamia and Asia Minor. Writings from the Ancient World 6. Atlanta, GA: Scholars Press, 1995.