I am trying to identify a pop song from around 1964, whose words
include: "Every time I see a girl like you, it makes me want to weep,
they remind me of you, they remind me of you -oo -oo -oo -oo, the
things that they do, they remind me of you.  Every time I see you
nightly, that night I just can't sleep".  Very Beatles type of song,
possibly even written by the Beatles for some other group but (I don't
think) recorded by them.

Dear 849fgt,

That song is "They Remind Me Of You", by the British beat band Billy
J. Kramer & The Dakotas.

The song was written by the band members Robin McDonald and Mike
Maxfield, and produced by George Martin. It was recorded on 23 January
1964 amd released in the UK on 21 February 1964 as the B-side of the
Parlophone single R5105. The A-side, "Little Children", reached #1 on
the British charts on 21 March 1964.

In the USA, "They Remind Me Of You" was released as the B-side of
"It's Gotta Last Forever" (Imperial 66085) on 8 June 1964.

In case you would like to buy a CD containing this song, you may be
interested in "Billy J. Kramer & The Dakots at Abbey Road, 1963-1966",
which is available from Amazon for $20.99:
http://www.amazon.com/exec/obidos/tg/detail/-/B000024XT0/

Hope this answers your question!
Regards,
Scriptor

Couldn;t ask for better.  Many thanks!