That’s an interesting period to compare, ostensibly the two corps were very alike, both were the elite landing forces of their respective countries and fulfilled similar roles. During the Boxer Rebellion in 1900 and the subsequent international relief campaign the two corps fought side by side.
Royal Marines
In 1900, the British Royal Marines were named the Royal Marines Light Infantry and as such were trained to act as a light infantry force which would be landed ahead of a main force to secure the immediate landing area. During the Boxer Rebellion the RMLI wore a dark navy blue jacket and trousers with either a field service cap or Wolseley helmet, by the time the First World War began the uniform had not much changed and the Royal Marines which fought at Antwerp in 1914 wore much the same uniform but wore the distinctive service Broderick cap.
RMLI c.1914 (source)
Until 1890, the Royal Marines were armed with the .577/450 Martini-Henry single shot breech loader. This had been in service since 1871 and had been used in dozens of colonial campaigns. The Martini-Henry was replaced by the Lee-Metford bolt action magazine rifle. The Lee-Metford was the precursor to the famous Lee-Enfield Rifle. The Lee-Metford itself was replaced in 1895 by the Magazine Lee-Enfield - much the same rifle except the rifling had been improved to handle smokeless powder.
Charger Loading Lee-Enfield (source)
By 1905, these MLEs had been converted to load from charger clips and redesignated Charger Loading Lee-Enfields (CLLEs). In 1907, the earlier Lee-Enfield rifle patterns were replaced with the SMLE which was significantly shorter and handier than the earlier rifles. You can read more about the evolution of the Lee-Enfield here. In 1914 there was a significant shortage of rifles as the British Army expanded rapidly and dried up all existing stocks of rifles. As such the marines and the Royal Naval Division which they formed with volunteers from the Royal Navy in 1914 were often armed with the older Charger Loading Lee-Enfields until production of SMLEs caught up.
As for side arms the British military of the period used a variety of Webley Revolvers. These were mainly used by officers (who often bought them privately) and by designated marines such as buglers, signalers and some NCOs.
British Vickers Machine Gun MKI (source)
The RMLI was armed much as the regular British Army was in 1914, the machine gun they were issued with was the improved Vickers gun adopted in 1912.
US Marine Corps
The US Marine Corps of the 19th century was very small in comparison with its current form, made up of a hand full of regiments. However, it came into its own during the US’ Banana Wars which involved US military action across the Caribbean and Pacific. The USMC was present during the Spanish-American War, the US’ interventions in Panama, Mexico, Nicaragua, the Philippine–American War and the Boxer Rebellion.
USMC c.1900 (source)
The USMC was uniformed much as the US Army was by 1900, with navy blue jacket and trousers and khaki campaign hat (see above). By 1914, the blue had been replaced by a khaki/sand coloured service dress for use on campaign. The marines who arrived in France in 1917 wore khaki tunics, trousers and M1912 Service hat (and Brodie helmet).
Illustration showing the evolution of the USMC’s uniform (source)
In 1900 the USMC was armed with the M1892-98 Krag rifle, a Norwegian design the US Krag was chambered in .30-40 Krag which had proved itself not fit for purpose during the Spanish-American War where it was outclassed by the Spanish Mausers it faced. However the rifle wasn’t fully withdrawn from service until around 1910.
M1892-98 Krag (source)
The Krag was replaced by the Springfield M1903, a Mauser derivative, chambered in the new .30-06 spitzer round. Despite a false start and some revisions the M1903 was widely issued by 1905 and many were carried during the Banana Wars and during World War One.
The sidearms used by the USMC during the period were quite varied, ranging from the M1905 and M1909 Colt revolvers chambered in .38 and .45, there was also some use of the venerable Colt Single Action Army when it was found the .38 Revolvers lacked stopping power. However, by 1912 the USMC had followed the US Army’s lead and adopted Colt M1911 semi-automatic pistol in .45ACP - which would remain in service for over 70 years.
General Barnett testing a Colt-Browning M1895, with a Lewis Gun in the foreground and a Hotchkiss M1909 in the background (source)
As for machine guns the USMC used a number of different weapons, initially using the Colt-Browning M1895 ‘potato-digger’ machine guns (adopted by the US Navy but not the Army) as well as the M1909 Benet-Mercie light machine gun which saw extensive service. These were replaced in time by the M1915 Vickers and later the M1917 Browning heavy machine guns, and the Lewis Light Machine Gun which was adopted by the USMC but not the US Army. The M1918 Browning Automatic Rifle and the Winchester M1897 Trench gun were also adopted during World War One.
I hope that adequately answers your question, thanks for the message