In terms of Heraldry, they are not at all similar. The form used is standard to most costs of arms (shield in middle featuring family blazons, supporting guardian animals on each side, helm with mantle and/or coronet above, motto below.)

The supporters of both crests happen to be the same; I don't know the Rothschild's heritage, but in the case of the United Kingdom, the Lion symbolizes England, while the Unicorn symbolizes Scotland. The British shield is quartered with the escutcheons of Ireland (harp), three lions for England and a single rampant lion for Scotland. They are quartered together in this way due to Heraldic conventions on depicting the lineage of the armiger, in this case the Queen of Great Britain. (Unlike, say, the Seal of the United States, which symbolizes the country, this coat of arms is personalized to the Monarch, and will change based upon the heritage of the ruler, altho practically speaking the UK Coat of Arms in its current form, dating from 1837, will probably continue unchanged for some time to come.)

One can "read" the history of an armiger and their family thru their Coat of Arms. Arms that have been quartered, like those of the Rothschilds and the UK, imply a union of two strong families, possible involving a heraldic heiress (a female child who is the sole heir of her family's arms.) Arms are passed down thru families just like inheritance and estates, and families who wish to play up an influential connection will often have their arms altered to reflect that. In the case of the Coat of Arms of the UK, it has been historically altered to reflect real-world political changes in the form of conquest or unification with other Sovereign countries like Scotland or Ireland.